%!PS-Adobe-2.0 %%Creator: dvips(k) 2020.1 Copyright 2020 Radical Eye Software %%Title: arXiv:2303.15010v1 [math.NT] 27 Mar 2023 %%CreationDate: Tue Mar 28 07:29:04 2023 %%Pages: 10 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %%BeginDefaults %%ViewingOrientation: 1 0 0 1 %%EndDefaults %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -R2 -P pk -z %+ p_adic_harmonic_sum_rev_cleaned.dvi -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2023.03.28:0729 %%BeginProcSet: texc.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /dir 0 def/dyy{/dir 0 def}B/dyt{/dir 1 def}B/dty{/dir 2 def}B/dtt{/dir 3 def}B/p{dir 2 eq{-90 rotate show 90 rotate}{dir 3 eq{-90 rotate show 90 rotate}{show}ifelse}ifelse}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)(LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse} forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{ BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat {BDot}imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B /M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M} B/g{0 M}B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{ 0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: psfrag.pro 0 0 %% %% This is file `psfrag.pro', %% generated with the docstrip utility. %% %% The original source files were: %% %% psfrag.dtx (with options: `filepro') %% %% Copyright (c) 1996 Craig Barratt, Michael C. 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y(2124679\),)21 b(whic)n(h)h(are)h(W)-7 b(olstenholme)20 b(primes)h(and)i(satisfy)f FB(\027)1986 394 y Fp(16843)2156 382 y Fs(\()p FB(H)7 b Fs(\(16842\)\))21 b(=)i(3)g(and)g FB(\027)2942 394 y Fp(2124678)3178 382 y Fs(\()p FB(H)7 b Fs(\(2124679\)\))20 b Fq(\025)0 482 y Fs(3.)36 b(Finally)-7 b(,)25 b(Bo)n(yd)i([)639 482 y SDict begin H.S end 639 482 a 0 TeXcolorgray Fs(3)p 0 TeXcolorgray 681 428 a SDict begin H.R end 681 428 a 681 482 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Boyd) cvn H.B /ANN pdfmark end 681 482 a -1 w Fs(])h(conjectured)f(that) h(there)g(are)e(no)i(pairs)d(\()p FB(n;)14 b(p)p Fs(\))28 b(suc)n(h)f(that)h FB(\027)2655 494 y Fo(p)2693 482 y Fs(\()p FB(H)7 b Fs(\()p FB(n)p Fs(\)\))24 b Fq(\025)f Fs(4.)0 582 y SDict begin H.S end 0 582 a 0 582 a SDict begin 12 H.A end 0 582 a 0 582 a SDict begin [/View [/XYZ H.V]/Dest (section.2) cvn /DEST pdfmark end 0 582 a 1664 658 a Fs(2.)41 b Fr(Resul)-6 b(ts)100 807 y Fs(In)27 b([)226 807 y SDict begin H.S end 226 807 a 0 TeXcolorgray Fs(3)p 0 TeXcolorgray 268 754 a SDict begin H.R end 268 754 a 268 807 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Boyd) cvn H.B /ANN pdfmark end 268 807 a Fs(])h(and)f([)503 807 y SDict begin H.S end 503 807 a 0 TeXcolorgray Fs(6)p 0 TeXcolorgray 545 754 a SDict begin H.R end 545 754 a 545 807 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.EL) cvn H.B /ANN pdfmark end 545 807 a Fs(])g(the)h(follo)n(wing)c (Lemma)h(has)i(b)r(een)h(pro)n(v)n(ed:)0 990 y SDict begin H.S end 0 990 a 0 990 a SDict begin 12 H.A end 0 990 a 0 990 a SDict begin [/View [/XYZ H.V]/Dest (Theorem.2.1) cvn /DEST pdfmark end 0 990 a 0 TeXcolorgray FA(Lemma)i(2.1.)p 0 TeXcolorgray 40 w Fh(F)-7 b(or)27 b(an)n(y)f(prime)g FB(p)d Fq(\025)f Fs(5)p Fh(,)28 b(if)f FB(\027)1432 1002 y Fo(p)1470 990 y Fs(\()p FB(H)7 b Fs(\()p FB(n)p Fs(\)\))24 b Fq(\024)e Fs(2)28 b Fh(then)1387 1127 y FB(\027)1428 1139 y Fo(p)1466 1127 y Fs(\()p FB(H)7 b Fs(\()p FB(pn)p Fs(\)\))24 b(=)f FB(\027)1915 1139 y Fo(p)1953 1127 y Fs(\()p FB(H)7 b Fs(\()p FB(n)p Fs(\)\))19 b Fq(\000)f Fs(1)p FB(:)100 1276 y Fs(Bo)n(yd)34 b([)346 1276 y SDict begin H.S end 346 1276 a 0 TeXcolorgray Fs(3)p 0 TeXcolorgray 388 1223 a SDict begin H.R end 388 1223 a 388 1276 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Boyd) cvn H.B /ANN pdfmark end 388 1276 a Fs(])h(giv)n(es)f(a)h(stronger)f(v)n (ersion)f(of)i(this)g(lemma,)g(whic)n(h)f(w)n(e)h(ha)n(v)n(e)g(p)r (ostp)r(oned)g(to)h(Prop)r(osition)3479 1276 y SDict begin H.S end 3479 1276 a 0 TeXcolorgray Fs(3.5)p 0 TeXcolorgray 3586 1223 a SDict begin H.R end 3586 1223 a 3586 1276 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.3.5) cvn H.B /ANN pdfmark end 3586 1276 a Fs(.)59 b(A)n(t)0 1376 y(this)28 b(in)n(tro)r(ductory)f(stage,)i(in)g(fact,)g(a)g(w)n (eak)n(er)e(v)n(ersion)g(is)h(su\016cien)n(t)g(to)h(explains)e(what)i (w)n(e)g(will)d(call)h(the)i(\\descen)n(t)0 1476 y(phenomenon")18 b(observ)n(ed)g(in)g(T)-7 b(able)1160 1476 y SDict begin H.S end 1160 1476 a 0 TeXcolorgray Fs(1)p 0 TeXcolorgray 1202 1422 a SDict begin H.R end 1202 1422 a 1202 1476 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (table.1) cvn H.B /ANN pdfmark end 1202 1476 a 20 w Fs(of)19 b(the)h FB(p)p Fs(-adic)d(v)-5 b(aluations)17 b(of)i(harmonic)e(n)n(um)n(b)r (ers.)33 b(Starting)18 b(from)g(that)h(p)r(oin)n(t,)0 1575 y(w)n(e)29 b(can)g(see)g(that)g(the)h(b)r(eha)n(viour)d(of)j (harmonic)c(n)n(um)n(b)r(ers)i(when)h FB(\027)2186 1587 y Fo(p)2225 1575 y Fs(\()p FB(H)7 b Fs(\()p FB(n)p Fs(\)\))26 b Fq(\024)g Fs(2)j(in)f(terms)g(of)i FB(p)p Fs(-adic)d(v)-5 b(aluation)27 b(is)0 1675 y(the)e(follo)n(wing:)31 b FB(\027)562 1687 y Fo(p)600 1675 y Fs(\()p FB(H)7 b Fs(\()p FB(p)782 1645 y Fo(m)845 1675 y FB(n)p Fs(\)\))24 b(=)e FB(\027)1111 1687 y Fo(p)1150 1675 y Fs(\()p FB(H)7 b Fs(\()p FB(n)p Fs(\)\))12 b Fq(\000)g FB(m)24 b Fs(for)g(all)e FB(m)h Fq(2)g Fi(N)t Fs(;)31 b(here)24 b(w)n(e)g(giv)n(e)e(a)i(fragmen) n(t)f(for)h(the)g(table)f(of)i FB(p)d Fs(=)h(5.)p 0 TeXcolorgray 0 1802 a SDict begin H.S end 0 1802 a 0 1802 a SDict begin 12 H.A end 0 1802 a 0 1802 a SDict begin [/View [/XYZ H.V]/Dest (table.1) cvn /DEST pdfmark end 0 1802 a 0 TeXcolorgray 299 1864 a Fr(T)-7 b(able)35 b(1.)694 1864 y SDict begin H.S end 694 1864 a 694 1864 a SDict begin H.R end 694 1864 a 694 1864 a SDict begin [/View [/XYZ H.V]/Dest (table.1) cvn /DEST pdfmark end 694 1864 a Fs(T)-7 b(able)30 b(for)h FB(p)d Fs(=)h(5:)44 b(this)30 b(sho)n(ws)g(the)i(5-adic)d(v)-5 b(aluation)29 b(of)i FB(H)7 b Fs(\(5)p FB(m)20 b Fs(+)h FB(k)s Fs(\),)32 b(where)f FB(m)g Fs(is)p 0 TeXcolorgray 299 1964 a(the)d(column)d (index)i(and)h FB(k)i Fs(is)d(the)h(ro)n(w)e(index.)p 901 2075 1959 4 v 901 2174 4 100 v 1608 2144 a FB(\027)1649 2156 y Fp(5)1686 2144 y Fs(\()p FB(H)7 b Fs(\(5)p FB(m)18 b Fs(+)g FB(k)s Fs(\)\))p 2856 2174 V 901 2178 1959 4 v 901 2277 4 100 v 1224 2277 V 1243 2277 V 1307 2247 a FB(k)26 b Fs(=)d(0)p 1566 2277 V 124 w FB(k)j Fs(=)d(1)p 1889 2277 V 124 w FB(k)i Fs(=)e(2)p 2211 2277 V 124 w FB(k)j Fs(=)d(3)p 2534 2277 V 124 w FB(k)j Fs(=)c(4)p 2856 2277 V 901 2281 1959 4 v 901 2301 V 901 2400 4 100 v 951 2370 a FB(m)h Fs(=)g(0)p 1224 2400 V 1243 2400 V 188 w Fq(1)p 1566 2400 V 260 w Fs(0)p 1889 2400 V 281 w(0)p 2211 2400 V 280 w(0)p 2534 2400 V 281 w(2)p 2856 2400 V 901 2500 V 1043 2470 a(1)p 1224 2500 V 1243 2500 V 287 w(-1)p 1566 2500 V 252 w(-1)p 1889 2500 V 253 w(-1)p 2211 2500 V 253 w(-1)p 2534 2500 V 252 w(-1)p 2856 2500 V 901 2599 V 1043 2570 a(2)p 1224 2599 V 1243 2599 V 287 w(-1)p 1566 2599 V 252 w(-1)p 1889 2599 V 253 w(-1)p 2211 2599 V 253 w(-1)p 2534 2599 V 252 w(-1)p 2856 2599 V 901 2699 V 1043 2669 a(3)p 1224 2699 V 1243 2699 V 287 w(-1)p 1566 2699 V 252 w(-1)p 1889 2699 V 253 w(-1)p 2211 2699 V 253 w(-1)p 2534 2699 V 252 w(-1)p 2856 2699 V 901 2799 V 1043 2769 a(4)p 1224 2799 V 1243 2799 V 301 w(1)p 1566 2799 V 280 w(0)p 1889 2799 V 281 w(0)p 2211 2799 V 280 w(0)p 2534 2799 V 281 w(1)p 2856 2799 V 901 2898 V 1043 2868 a(5)p 1224 2898 V 1243 2898 V 287 w(-2)p 1566 2898 V 252 w(-2)p 1889 2898 V 253 w(-2)p 2211 2898 V 253 w(-2)p 2534 2898 V 252 w(-2)p 2856 2898 V 901 2998 V 1043 2968 a(6)p 1224 2998 V 1243 2998 V 287 w(-2)p 1566 2998 V 252 w(-2)p 1889 2998 V 253 w(-2)p 2211 2998 V 253 w(-2)p 2534 2998 V 252 w(-2)p 2856 2998 V 901 3153 4 155 v 1052 3057 a(.)1052 3090 y(.)1052 3123 y(.)p 1224 3153 V 1243 3153 V 1395 3057 a(.)1395 3090 y(.)1395 3123 y(.)p 1566 3153 V 1717 3057 a(.)1717 3090 y(.)1717 3123 y(.)p 1889 3153 V 2040 3057 a(.)2040 3090 y(.)2040 3123 y(.)p 2211 3153 V 2363 3057 a(.)2363 3090 y(.)2363 3123 y(.)p 2534 3153 V 2685 3057 a(.)2685 3090 y(.)2685 3123 y(.)p 2856 3153 V 901 3252 4 100 v 1022 3223 a(19)p 1224 3252 V 1243 3252 V 266 w(-2)p 1566 3252 V 252 w(-2)p 1889 3252 V 253 w(-2)p 2211 3252 V 253 w(-2)p 2534 3252 V 252 w(-2)p 2856 3252 V 901 3352 V 1022 3322 a(20)p 1224 3352 V 1243 3352 V 280 w(0)p 1566 3352 V 280 w(0)p 1889 3352 V 281 w(0)p 2211 3352 V 280 w(0)p 2534 3352 V 281 w(0)p 2856 3352 V 901 3452 V 1022 3422 a(21)p 1224 3452 V 1243 3452 V 266 w(-1)p 1566 3452 V 252 w(-1)p 1889 3452 V 253 w(-1)p 2211 3452 V 253 w(-1)p 2534 3452 V 252 w(-1)p 2856 3452 V 901 3551 V 1022 3521 a(22)p 1224 3551 V 1243 3551 V 266 w(-1)p 1566 3551 V 252 w(-1)p 1889 3551 V 253 w(-1)p 2211 3551 V 253 w(-1)p 2534 3551 V 252 w(-1)p 2856 3551 V 901 3651 V 1022 3621 a(23)p 1224 3651 V 1243 3651 V 266 w(-1)p 1566 3651 V 252 w(-1)p 1889 3651 V 253 w(-1)p 2211 3651 V 253 w(-1)p 2534 3651 V 252 w(-1)p 2856 3651 V 901 3751 V 1022 3721 a(24)p 1224 3751 V 1243 3751 V 280 w(0)p 1566 3751 V 280 w(0)p 1889 3751 V 281 w(0)p 2211 3751 V 280 w(0)p 2534 3751 V 281 w(0)p 2856 3751 V 901 3850 V 1022 3820 a(25)p 1224 3850 V 1243 3850 V 266 w(-3)p 1566 3850 V 252 w(-3)p 1889 3850 V 253 w(-3)p 2211 3850 V 253 w(-3)p 2534 3850 V 252 w(-3)p 2856 3850 V 901 4005 4 155 v 1052 3909 a(.)1052 3942 y(.)1052 3975 y(.)p 1224 4005 V 1243 4005 V 1395 3909 a(.)1395 3942 y(.)1395 3975 y(.)p 1566 4005 V 1717 3909 a(.)1717 3942 y(.)1717 3975 y(.)p 1889 4005 V 2040 3909 a(.)2040 3942 y(.)2040 3975 y(.)p 2211 4005 V 2363 3909 a(.)2363 3942 y(.)2363 3975 y(.)p 2534 4005 V 2685 3909 a(.)2685 3942 y(.)2685 3975 y(.)p 2856 4005 V 901 4105 4 100 v 1002 4075 a(124)p 1224 4105 V 1243 4105 V 244 w(-1)p 1566 4105 V 252 w(-1)p 1889 4105 V 253 w(-1)p 2211 4105 V 253 w(-1)p 2534 4105 V 252 w(-1)p 2856 4105 V 901 4204 V 1002 4174 a(125)p 1224 4204 V 1243 4204 V 244 w(-4)p 1566 4204 V 252 w(-4)p 1889 4204 V 253 w(-4)p 2211 4204 V 253 w(-4)p 2534 4204 V 252 w(-4)p 2856 4204 V 901 4304 V 1002 4274 a(126)p 1224 4304 V 1243 4304 V 244 w(-4)p 1566 4304 V 252 w(-4)p 1889 4304 V 253 w(-4)p 2211 4304 V 253 w(-4)p 2534 4304 V 252 w(-4)p 2856 4304 V 901 4459 4 155 v 1052 4363 a(.)1052 4396 y(.)1052 4429 y(.)p 1224 4459 V 1243 4459 V 1395 4363 a(.)1395 4396 y(.)1395 4429 y(.)p 1566 4459 V 1717 4363 a(.)1717 4396 y(.)1717 4429 y(.)p 1889 4459 V 2040 4363 a(.)2040 4396 y(.)2040 4429 y(.)p 2211 4459 V 2363 4363 a(.)2363 4396 y(.)2363 4429 y(.)p 2534 4459 V 2685 4363 a(.)2685 4396 y(.)2685 4429 y(.)p 2856 4459 V 901 4462 1959 4 v 0 TeXcolorgray 100 4641 a(Ho)n(w)n(ev)n(er,)35 b(Lemma)771 4641 y SDict begin H.S end 771 4641 a 0 TeXcolorgray Fs(2.1)p 0 TeXcolorgray 878 4588 a SDict begin H.R end 878 4588 a 878 4641 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.2.1) cvn H.B /ANN pdfmark end 878 4641 a 34 w Fs(cannot)g(b)r(e)h(applied)d(to)i(the)h(case)f FB(\027)2096 4653 y Fo(p)2134 4641 y Fs(\()p FB(H)7 b Fs(\()p FB(n)p Fs(\)\))37 b Fq(\025)e Fs(3.)60 b(W)-7 b(e)35 b(pro)n(vide)e(form)n(ulas)g(that)i(can)0 4741 y(apply)26 b(to)i(this)f(case)f(as)h(w)n(ell.)0 4923 y SDict begin H.S end 0 4923 a 0 4923 a SDict begin 12 H.A end 0 4923 a 0 4923 a SDict begin [/View [/XYZ H.V]/Dest (Theorem.2.2) cvn /DEST pdfmark end 0 4923 a 0 TeXcolorgray FA(De\014nition)k(2.2.)p 0 TeXcolorgray 40 w Fh(The)d(Bernoulli)23 b(n)n(um)n(b)r(ers)j FB(B)1563 4935 y Fo(n)1636 4923 y Fh(are)h(de\014ned)h(using)e(the)i(generating)d(function)1634 5066 y FB(t)p 1543 5103 211 4 v 1543 5179 a Fs(1)18 b Fq(\000)g FB(e)1725 5155 y Fo(t)1787 5122 y Fs(=)1902 5018 y Fk(1)1875 5043 y Fl(X)1875 5222 y Fo(k)q Fp(=0)2010 5122 y FB(B)2073 5134 y Fo(k)2123 5066 y FB(t)2153 5036 y Fo(k)p 2123 5103 71 4 v 2124 5179 a FB(k)s Fs(!)2204 5122 y FB(:)0 5384 y SDict begin H.S end 0 5384 a 0 5384 a SDict begin 12 H.A end 0 5384 a 0 5384 a SDict begin [/View [/XYZ H.V]/Dest (Theorem.2.3) cvn /DEST pdfmark end 0 5384 a 0 TeXcolorgray FA(De\014nition)31 b(2.3.)p 0 TeXcolorgray 40 w Fh(A)d(prime)d FB(p)j Fh(is)e(a)h(W)-7 b(olstenholme)25 b(prime)g(if)i FB(p)c Fq(j)g FB(B)2204 5396 y Fo(p)p Fk(\000)p Fp(3)2328 5384 y Fh(.)0 5567 y SDict begin H.S end 0 5567 a 0 5567 a SDict begin 12 H.A end 0 5567 a 0 5567 a SDict begin [/View [/XYZ H.V]/Dest (Theorem.2.4) cvn /DEST pdfmark end 0 5567 a 0 TeXcolorgray FA(Prop)s(osition)30 b(2.4.)p 0 TeXcolorgray 40 w FB(p)e Fh(is)e(a)h(W)-7 b(olstenholme)25 b(prime)g(if)i(and)h(only)e(if)h FB(p)2237 5537 y Fp(3)2297 5567 y Fq(j)c FB(H)7 b Fs(\()p FB(p)19 b Fq(\000)f Fs(1\))p Fh(.)0 5716 y Fs(This)28 b(result)g(explains)e(the)k(fact)f(that)g(W)-7 b(olstenholme)26 b(primes)h(giv)n(e)g(o)r(ccurrences)h(of)h FB(\027)2801 5728 y Fo(p)2839 5716 y Fs(\()p FB(H)7 b Fs(\()p FB(n)p Fs(\)\))26 b(=)f(3)k(for)f FB(n)e Fs(=)f FB(p)19 b Fq(\000)g Fs(1)0 5816 y(as)32 b(in)h(the)g(in)n(tro)r(duction.)52 b(It)33 b(is)f(imp)r(ortan)n(t)f(to)i(rep)r(ort)f(this)h(result)f (\(whic)n(h)g(comes)g(from)f(a)i(pap)r(er)g(submitted)f(b)n(y)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 3 3 TeXDict begin HPSdict begin 3 2 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -8 -191 a SDict begin H.S end -8 -191 a -8 -191 a SDict begin H.R end -8 -191 a -8 -191 a SDict begin [/View [/XYZ H.V]/Dest (page.3) cvn /DEST pdfmark end -8 -191 a 0 TeXcolorgray 155 -116 a Fo(p)p Fp(-ADIC)30 b(V)-7 b(ALUA)i(TION)29 b(OF)g(HARMONIC)h(SUMS)e(AND)h(THEIR)g (CONNECTIONS)h(WITH)f(W)n(OLSTENHOLME)f(PRIMES)122 b(3)p 0 TeXcolorgray 0 83 a Fs(Gardiner)17 b([)366 83 y SDict begin H.S end 366 83 a 0 TeXcolorgray Fs(8)p 0 TeXcolorgray 408 30 a SDict begin H.R end 408 30 a 408 83 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.gardiner) cvn H.B /ANN pdfmark end 408 83 a Fs(])j(in)e(1988\),)i(as)f(it)g(sho)n(ws)f(also) g(other)h(equiv)-5 b(alen)n(t)17 b(de\014nitions)h(of)h(W)-7 b(ohstenholme)17 b(primes,)i(that,)j(ho)n(w)n(ev)n(er,)0 183 y(w)n(e)27 b(did)g(not)h(use)f(in)g(this)g(article.)100 282 y(Belo)n(w)e(w)n(e)i(presen)n(t)g(the)h(main)e(results)g(of)i(our)e (researc)n(h:)0 465 y SDict begin H.S end 0 465 a 0 465 a SDict begin 12 H.A end 0 465 a 0 465 a SDict begin [/View [/XYZ H.V]/Dest (Theorem.2.5) cvn /DEST pdfmark end 0 465 a 0 TeXcolorgray FA(Theorem)33 b(2.5.)p 0 TeXcolorgray 40 w Fh(Let)d FB(p)c Fq(\025)g Fs(5)j Fh(b)r(e)h(a)f(non-W)-7 b(olstenholme)26 b(prime)i(and)h FB(n)d Fq(2)h Fi(N)40 b Fh(suc)n(h)29 b(that)h FB(p)c Fi(-)h FB(n)i Fh(and)h FB(\027)3286 477 y Fo(p)3324 465 y Fs(\()p FB(H)7 b Fs(\()p FB(n)p Fs(\)\))27 b Fq(\025)f Fs(3)p Fh(.)0 565 y(Then)0 591 y SDict begin H.S end 0 591 a 0 591 a SDict begin 12 H.A end 0 591 a 0 591 a SDict begin [/View [/XYZ H.V]/Dest (Item.1) cvn /DEST pdfmark end 0 591 a 0 TeXcolorgray 141 690 a Fh(\(1\))p 0 TeXcolorgray 42 w(If)i FB(\027)413 702 y Fo(p)451 690 y Fs(\()p FB(H)7 b Fs(\()p FB(n)p Fs(\)\))24 b Fq(\025)f Fs(4)p Fh(,)k(then)h FB(\027)1139 702 y Fo(p)1177 690 y Fs(\()p FB(H)7 b Fs(\()p FB(pn)p Fs(\)\))24 b(=)f(2)k Fh(and)g(then)h(w)n(e)f(p)r(erceiv)n(e)f(the)i (\\descen)n(t)f(phenomenon",)0 714 y SDict begin H.S end 0 714 a 0 714 a SDict begin 12 H.A end 0 714 a 0 714 a SDict begin [/View [/XYZ H.V]/Dest (Item.2) cvn /DEST pdfmark end 0 714 a 0 TeXcolorgray 141 790 a Fh(\(2\))p 0 TeXcolorgray 42 w(If)d FB(\027)409 802 y Fo(p)447 790 y Fs(\()p FB(H)7 b Fs(\()p FB(n)p Fs(\)\))24 b(=)e(3)h Fh(and)h(there)f(do)r(es)g(not)g(exists)f FB(m)h Fq(2)h Fi(N)33 b Fh(suc)n(h)23 b(that)h FB(\027)2452 802 y Fo(p)2490 790 y Fs(\()p FB(H)7 b Fs(\()p FB(p)2672 760 y Fo(m)2735 790 y FB(n)p Fs(\)\))24 b Fq(\024)e Fs(2)p Fh(,)i(then)g FB(\027)3275 802 y Fo(p)3313 790 y Fs(\()p FB(H)7 b Fs(\()p FB(p)3495 760 y Fo(m)3559 790 y FB(n)p Fs(\)\))23 b(=)289 889 y(2)p FB(m)18 b Fs(+)g(3)27 b Fh(for)g(all)e FB(m)e Fq(2)h Fi(N)s Fh(,)0 906 y SDict begin H.S end 0 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1269 90 4 v 2398 1345 a Fs(3)2492 1288 y(+)18 b(1)1674 1483 y FB(M)27 b Fq(\000)19 b FB(m)165 b Fh(if)28 b FB(m)23 b(>)2374 1426 y(M)p 2374 1463 V 2398 1539 a Fs(3)2492 1483 y(+)18 b(1)289 1657 y Fh(for)27 b(all)e FB(m)e Fq(2)g Fi(N)t Fh(.)0 1840 y SDict begin H.S end 0 1840 a 0 1840 a SDict begin 12 H.A end 0 1840 a 0 1840 a SDict begin [/View [/XYZ H.V]/Dest (Theorem.2.6) cvn /DEST pdfmark end 0 1840 a 0 TeXcolorgray FA(Theorem)k(2.6.)p 0 TeXcolorgray 40 w Fh(Let)f FB(p)c Fq(\025)h Fs(5)i Fh(b)r(e)g(a)g(W)-7 b(olstenholme)22 b(prime,)h(and)i FB(n)e Fq(2)g Fi(N)35 b Fh(suc)n(h)25 b(that)g FB(p)e Fi(-)g FB(n)i Fh(and)g FB(\027)3067 1852 y Fo(p)3105 1840 y Fs(\()p FB(H)7 b Fs(\()p FB(n)p Fs(\)\))24 b Fq(\025)f Fs(3)p Fh(.)36 b(Then)0 1890 y SDict begin H.S end 0 1890 a 0 1890 a SDict begin 12 H.A end 0 1890 a 0 1890 a SDict begin [/View [/XYZ H.V]/Dest (Item.4) cvn /DEST pdfmark end 0 1890 a 0 TeXcolorgray 141 1966 a Fh(\(1\))p 0 TeXcolorgray 42 w(If)28 b FB(\027)413 1978 y Fo(p)451 1966 y Fs(\()p FB(H)7 b Fs(\()p FB(n)p Fs(\)\))24 b(=)f(3)p Fh(,)k(then)h FB(\027)1139 1978 y Fo(p)1177 1966 y Fs(\()p FB(H)7 b Fs(\()p FB(pn)p Fs(\)\))24 b(=)f(2)k Fh(and)g(then)h(w)n(e)f(p)r (erceiv)n(e)f(the)i(\\descen)n(t)f(phenomenon",)0 1989 y SDict begin H.S end 0 1989 a 0 1989 a SDict begin 12 H.A end 0 1989 a 0 1989 a SDict begin [/View [/XYZ H.V]/Dest (Item.5) cvn /DEST pdfmark end 0 1989 a 0 TeXcolorgray 141 2065 a Fh(\(2\))p 0 TeXcolorgray 42 w(If)d FB(\027)409 2077 y Fo(p)447 2065 y Fs(\()p FB(H)7 b Fs(\()p FB(n)p Fs(\)\))24 b Fq(\025)e Fs(4)h Fh(and)h(there)f(do)r(es)g(not)g(exists)f FB(m)h Fq(2)h Fi(N)33 b Fh(suc)n(h)23 b(that)h FB(\027)2452 2077 y Fo(p)2490 2065 y Fs(\()p FB(H)7 b Fs(\()p FB(p)2672 2035 y Fo(m)2735 2065 y FB(n)p Fs(\)\))24 b Fq(\024)e Fs(3)p Fh(,)i(then)g FB(\027)3275 2077 y Fo(p)3313 2065 y Fs(\()p FB(H)7 b Fs(\()p FB(p)3495 2035 y Fo(m)3559 2065 y FB(n)p Fs(\)\))23 b Fq(\025)289 2165 y Fs(2)p FB(m)18 b Fs(+)g(4)27 b Fh(for)g(all)e FB(m)e Fq(2)h Fi(N)s Fh(,)0 2181 y SDict begin H.S end 0 2181 a 0 2181 a SDict begin 12 H.A end 0 2181 a 0 2181 a SDict begin [/View [/XYZ H.V]/Dest (Item.6) cvn /DEST pdfmark end 0 2181 a 0 TeXcolorgray 141 2264 a Fh(\(3\))p 0 TeXcolorgray 42 w(If)32 b FB(\027)417 2276 y Fo(p)456 2264 y Fs(\()p FB(H)7 b Fs(\()p FB(n)p Fs(\)\))31 b Fq(\025)g Fs(4)h Fh(and)g(there)g(exists)f FB(m)f Fq(2)h Fi(N)43 b Fh(suc)n(h)32 b(that)g FB(\027)2227 2276 y Fo(p)2266 2264 y Fs(\()p FB(H)7 b Fs(\()p FB(p)2448 2234 y Fo(m)2511 2264 y FB(n)p Fs(\)\))31 b Fq(\024)f Fs(3)p Fh(,)j(then)g(there)f(exists)f FB(M)40 b Fq(2)31 b Fi(N)289 2366 y Fh(suc)n(h)c(that)h FB(\027)697 2378 y Fo(p)735 2366 y Fs(\()p FB(H)7 b Fs(\()p FB(p)917 2336 y Fo(M)991 2366 y FB(n)p Fs(\)\))24 b(=)e(0)27 b Fh(and)980 2458 y Fl(8)980 2533 y(>)980 2558 y(<)980 2707 y(>)980 2732 y(:)1053 2563 y FB(\027)1094 2575 y Fo(p)1133 2563 y Fs(\()p FB(H)7 b Fs(\()p FB(p)1315 2529 y Fo(m)1378 2563 y FB(n)p Fs(\)\))23 b Fq(\025)g Fs(2)p FB(m)18 b Fs(+)g(4)172 b Fh(if)27 b FB(m)c Fq(\024)2303 2507 y FB(M)k Fs(+)18 b(1)p 2303 2544 233 4 v 2398 2620 a(3)2564 2563 y(+)g(1)1053 2758 y FB(\027)1094 2770 y Fo(p)1133 2758 y Fs(\()p FB(H)7 b Fs(\()p FB(p)1315 2724 y Fo(m)1378 2758 y FB(n)p Fs(\)\))23 b(=)g FB(M)k Fq(\000)18 b FB(m)166 b Fh(if)27 b FB(m)c(>)2303 2702 y(M)k Fs(+)18 b(1)p 2303 2739 V 2398 2815 a(3)2564 2758 y(+)g(1)289 2933 y Fh(for)27 b(all)e FB(m)e Fq(2)g Fi(N)t Fh(.)0 3054 y SDict begin H.S end 0 3054 a 0 3054 a SDict begin 12 H.A end 0 3054 a 0 3054 a SDict begin [/View [/XYZ H.V]/Dest (section.3) cvn /DEST pdfmark end 0 3054 a 1387 3151 a Fs(3.)41 b Fr(Setting)32 b(the)g(pr)n(oblem)100 3300 y Fs(In)27 b(this)g(section)f(w)n(e)i(will)d(use)i(the)h(Big-O)d (notation,)1182 3452 y FB(\027)1223 3464 y Fo(p)1261 3452 y Fs(\()p FB(x)20 b Fq(\000)e FB(y)s Fs(\))23 b Fq(\025)f FB(k)109 b Fq(\()-14 b(\))106 b FB(x)19 b Fq(\000)f FB(y)26 b Fs(=)d FB(O)r Fs(\()p FB(p)2482 3418 y Fo(k)2523 3452 y Fs(\))p FB(:)0 3634 y SDict begin H.S end 0 3634 a 0 3634 a SDict begin 12 H.A end 0 3634 a 0 3634 a SDict begin [/View [/XYZ H.V]/Dest (Theorem.3.1) cvn /DEST pdfmark end 0 3634 a 0 TeXcolorgray FA(Prop)s(osition)30 b(3.1.)p 0 TeXcolorgray 40 w Fh(Let)e FB(a;)14 b(b)22 b Fq(2)i Fi(Z)o Fh(,)e Fs(\()p FB(b;)14 b(p)p Fs(\))23 b(=)f(1)p Fh(.)37 b(Then)1543 3751 y FB(a)p 1543 3788 44 4 v 1547 3864 a(b)1620 3807 y Fs(=)1809 3751 y FB(a)p 1717 3788 227 4 v 1717 3864 a(b)19 b Fs(+)f FB(hp)1973 3807 y Fs(+)g FB(O)r Fs(\()p FB(p)p Fs(\))0 3998 y Fh(for)27 b(all)e FB(h)e Fq(2)h Fi(Z)o Fh(.)p 0 TeXcolorgray 0 4148 a Fg(Pr)l(o)l(of.)p 0 TeXcolorgray 43 w Fs(W)-7 b(e)28 b(ha)n(v)n(e)1143 4310 y FB(\027)1184 4322 y Fo(p)1236 4193 y Fl(\022)1307 4253 y FB(a)p 1307 4291 44 4 v 1311 4367 a(b)1379 4310 y Fq(\000)1564 4253 y FB(a)p 1472 4291 227 4 v 1472 4367 a(b)18 b Fs(+)g FB(hp)1709 4193 y Fl(\023)1793 4310 y Fs(=)23 b FB(\027)1922 4322 y Fo(p)1974 4193 y Fl(\022)2142 4253 y FB(ahp)p 2045 4291 327 4 v 2045 4367 a(b)p Fs(\()p FB(b)18 b Fs(+)g FB(hp)p Fs(\))2382 4193 y Fl(\023)2466 4310 y Fq(\025)23 b Fs(1)p FB(;)3696 4487 y Ff(\003)0 4670 y SDict begin H.S end 0 4670 a 0 4670 a SDict begin 12 H.A end 0 4670 a 0 4670 a SDict begin [/View [/XYZ H.V]/Dest (Theorem.3.2) cvn /DEST pdfmark end 0 4670 a 0 TeXcolorgray FA(Prop)s(osition)h(3.2.)p 0 TeXcolorgray 40 w Fh(Let)f FB(q)j Fq(2)d Fi(Q)6 b Fh(.)41 b(Then)22 b FB(\027)1360 4682 y Fo(p)1399 4670 y Fs(\()p FB(q)s Fs(\))h(=)g FB(k)j(<)d Fs(1)f Fh(if)f(and)h(only)f(if)h(there)g (exist)g FB(a;)14 b(b)22 b Fq(2)h Fi(Z)p Fh(,)17 b FB(b)23 b Fq(6)p Fs(=)g(0)e Fh(with)h Fs(\()p FB(a;)14 b(p)p Fs(\))23 b(=)g(1)0 4770 y Fh(and)k Fs(\()p FB(b;)14 b(p)p Fs(\))23 b(=)g(1)k Fh(suc)n(h)g(that)1583 4908 y FB(q)g Fs(=)22 b FB(p)1776 4873 y Fo(k)1827 4851 y FB(a)p 1827 4888 44 4 v 1831 4965 a(b)1899 4908 y Fs(+)c FB(O)r Fs(\()p FB(p)p Fs(\))p FB(:)p 0 TeXcolorgray 0 5085 a Fg(Pr)l(o)l(of.)p 0 TeXcolorgray 0 TeXcolorgray 92 w Fs(\(=)-14 b Fq(\))p Fs(\))p 0 TeXcolorgray 42 w(Straigh)n(tforw)n(ard.)p 0 TeXcolorgray 49 5185 a(\()p Fq(\()g Fs(=\))p 0 TeXcolorgray 42 w(W)-7 b(e)28 b(ha)n(v)n(e)1553 5326 y FB(\027)1594 5338 y Fo(p)1646 5234 y Fl(\020)1695 5326 y FB(q)22 b Fq(\000)c FB(p)1879 5292 y Fo(k)1929 5270 y FB(a)p 1929 5307 V 1933 5383 a(b)1983 5234 y Fl(\021)2056 5326 y Fq(\025)23 b Fs(1)p FB(;)289 5483 y Fs(then)1077 5604 y FB(\027)1118 5616 y Fo(p)1156 5604 y Fs(\()p FB(q)s Fs(\))h(=)e(min)1524 5512 y Fl(n)1579 5604 y FB(\027)1620 5616 y Fo(p)1672 5512 y Fl(\020)1722 5604 y FB(p)1764 5570 y Fo(k)1814 5548 y FB(a)p 1814 5585 V 1818 5661 a(b)1868 5512 y Fl(\021)1932 5604 y FB(;)14 b(\027)2010 5616 y Fo(p)2062 5512 y Fl(\020)2111 5604 y FB(q)22 b Fq(\000)c FB(p)2295 5570 y Fo(k)2345 5548 y FB(a)p 2345 5585 V 2349 5661 a(b)2399 5512 y Fl(\021o)2527 5604 y Fs(=)23 b FB(k)s(:)3696 5642 y Ff(\003)100 5816 y Fs(No)n(w,)k(in)g (order)f(to)h(study)h(the)g(b)r(eha)n(viour)d(of)j FB(J)8 b Fs(\()p FB(p)p Fs(\),)28 b(w)n(e)f(m)n(ust)g(consider)f(a)h(v)n(ery)f (imp)r(ortan)n(t)g(prop)r(ert)n(y:)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 4 4 TeXDict begin HPSdict begin 4 3 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -8 -191 a SDict begin H.S end -8 -191 a -8 -191 a SDict begin H.R end -8 -191 a -8 -191 a SDict begin [/View [/XYZ H.V]/Dest (page.4) cvn /DEST pdfmark end -8 -191 a 0 TeXcolorgray 0 -116 a Fp(4)733 b(LEONARDO)23 b(CAR)n(OFIGLIO,)g(LUIGI)f(DE)g(FILPO,)g(ALESSANDR)n(O)g(GAMBINI)p 0 TeXcolorgray 0 83 a SDict begin H.S end 0 83 a 0 83 a SDict begin 12 H.A end 0 83 a 0 83 a SDict begin [/View [/XYZ H.V]/Dest (Theorem.3.3) cvn /DEST pdfmark end 0 83 a 0 TeXcolorgray FA(Lemma)30 b(3.3.)p 0 TeXcolorgray 40 w Fh(If)e FB(n)22 b Fq(2)i FB(J)8 b Fs(\()p FB(p)p Fs(\))p Fh(,)28 b FB(\027)1015 95 y Fo(p)1053 83 y Fs(\()p FB(H)7 b Fs(\()p FB(pn)p Fs(\)\))24 b Fq(\025)f Fs(0)p Fh(,)k(w)n(e)g(ha)n(v)n(e)1250 269 y FB(H)7 b Fs(\()p FB(pn)18 b Fs(+)g FB(k)s Fs(\))23 b(=)1750 213 y FB(H)7 b Fs(\()p FB(n)p Fs(\))p 1750 250 191 4 v 1824 326 a FB(p)1968 269 y Fs(+)18 b FB(H)7 b Fs(\()p FB(k)s Fs(\))19 b(+)f FB(O)r Fs(\()p FB(p)p Fs(\))0 443 y Fh(for)27 b(eac)n(h)g FB(k)f Fs(=)c(1)p FB(;)14 b(:)g(:)g(:)g(p)k Fq(\000)g Fs(1)p Fh(.)p 0 TeXcolorgray 0 593 a Fg(Pr)l(o)l(of.)p 0 TeXcolorgray 43 w Fs(Using)27 b(Lemma)790 593 y SDict begin H.S end 790 593 a 0 TeXcolorgray Fs(2.1)p 0 TeXcolorgray 897 539 a SDict begin H.R end 897 539 a 897 593 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.2.1) cvn H.B /ANN pdfmark end 897 593 a 27 w Fs(and)g(Prop)r(osition)1532 593 y SDict begin H.S end 1532 593 a 0 TeXcolorgray Fs(3.1)p 0 TeXcolorgray 1639 539 a SDict begin H.R end 1639 539 a 1639 593 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.3.1) cvn H.B /ANN pdfmark end 1639 593 a 27 w Fs(w)n(e)g(get)383 805 y FB(H)7 b Fs(\()p FB(pn)18 b Fs(+)g FB(k)s Fs(\))23 b(=)g FB(H)7 b Fs(\()p FB(pn)p Fs(\))18 b(+)1248 701 y Fo(k)1206 726 y Fl(X)1213 903 y Fo(i)p Fp(=1)1440 749 y Fs(1)p 1350 786 222 4 v 1350 862 a FB(pn)g Fs(+)g FB(i)1605 805 y Fs(=)1702 749 y FB(H)7 b Fs(\()p FB(n)p Fs(\))p 1702 786 191 4 v 1777 862 a FB(p)1921 805 y Fs(+)2046 701 y Fo(k)2004 726 y Fl(X)2010 903 y Fo(i)p Fp(=1)2148 749 y Fs(1)p 2148 786 42 4 v 2154 862 a FB(i)2218 805 y Fs(+)18 b FB(O)r Fs(\()p FB(p)p Fs(\))24 b(=)2593 749 y FB(H)7 b Fs(\()p FB(n)p Fs(\))p 2593 786 191 4 v 2668 862 a FB(p)2812 805 y Fs(+)18 b FB(H)7 b Fs(\()p FB(k)s Fs(\))19 b(+)f FB(O)r Fs(\()p FB(p)p Fs(\))p FB(;)3696 1011 y Ff(\003)100 1161 y Fs(F)-7 b(rom)26 b(Lemma)614 1161 y SDict begin H.S end 614 1161 a 0 TeXcolorgray Fs(3.3)p 0 TeXcolorgray 721 1108 a SDict begin H.R end 721 1108 a 721 1161 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.3.3) cvn H.B /ANN pdfmark end 721 1161 a 27 w Fs(follo)n(ws)e (immediately)f(the)28 b(follo)n(wing:)0 1344 y SDict begin H.S end 0 1344 a 0 1344 a SDict begin 12 H.A end 0 1344 a 0 1344 a SDict begin [/View [/XYZ H.V]/Dest (Theorem.3.4) cvn /DEST pdfmark end 0 1344 a 0 TeXcolorgray FA(Corollary)k(3.4.)p 0 TeXcolorgray 40 w Fh(If)c FB(\027)739 1356 y Fo(p)777 1344 y Fs(\()p FB(H)7 b Fs(\()p FB(pn)p Fs(\)\))24 b Fq(\025)e Fs(1)p Fh(,)28 b FB(H)7 b Fs(\()p FB(pn)18 b Fs(+)g FB(k)s Fs(\))23 b(=)g FB(H)7 b Fs(\()p FB(k)s Fs(\))18 b(+)g FB(O)r Fs(\()p FB(p)p Fs(\))p Fh(.)100 1493 y Fs(It)31 b(means)f(that)h(the)h(set)f(of)g(indices)f FB(k)h Fs(=)e(1)p FB(;)14 b(:)g(:)g(:)f(p)21 b Fq(\000)f Fs(1)31 b(suc)n(h)g(that)g FB(\027)2282 1505 y Fo(p)2321 1493 y Fs(\()p FB(H)7 b Fs(\()p FB(pn)20 b Fs(+)h FB(k)s Fs(\)\))29 b(=)g(0)i(coincides)d(with)j(the)h(set)0 1593 y(of)d(indices)d FB(k)i Fs(=)d(1)p FB(;)14 b(:)g(:)g(:)f(p)19 b Fq(\000)g Fs(1)28 b(as)g(b)r(efore)h(suc)n(h)f(that)h FB(\027)1698 1605 y Fo(p)1737 1593 y Fs(\()p FB(H)7 b Fs(\()p FB(k)s Fs(\)\))25 b(=)g(0.)40 b(This)28 b(explains)e(the)j(rep) r(etition)e(with)h(the)h(same)0 1692 y(patterns)i(of)g(zeros)f(observ)n (ed)g(in)g(T)-7 b(able)1320 1692 y SDict begin H.S end 1320 1692 a 0 TeXcolorgray Fs(1)p 0 TeXcolorgray 1362 1639 a SDict begin H.R end 1362 1639 a 1362 1692 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (table.1) cvn H.B /ANN pdfmark end 1362 1692 a Fs(,)32 b(ho)n(w)n(ev)n(er)d(w)n(e)i(cannot)g(y)n(et)g (predict)f(the)i(actual)e FB(p)p Fs(-adic)f(v)-5 b(aluation,)30 b(only)0 1792 y(whether)d(it)h(is)e(n)n(ull)g(or)g(not.)37 b(If,)28 b(on)g(the)g(other)f(hand,)g FB(\027)1783 1804 y Fo(p)1822 1792 y Fs(\()p FB(H)7 b Fs(\()p FB(pn)p Fs(\)\))23 b(=)g(0,)k(then)h(giv)n(en)e(that)1386 1978 y FB(H)7 b Fs(\()p FB(pn)19 b Fs(+)f FB(k)s Fs(\))23 b(=)1886 1922 y FB(H)7 b Fs(\()p FB(n)p Fs(\))p 1886 1959 191 4 v 1961 2035 a FB(p)2105 1978 y Fs(+)18 b FB(H)7 b Fs(\()p FB(k)s Fs(\))0 2157 y(for)29 b(eac)n(h)f FB(k)h Fs(=)c(1)p FB(;)14 b(:)g(:)g(:)f(p)20 b Fq(\000)f Fs(1,)29 b(the)h(study)f(of)h (these)f FB(k)j Fs(with)d FB(H)7 b Fs(\()p FB(pn)19 b Fs(+)g FB(k)s Fs(\))26 b(=)g(0)i(is)h(reduced)f(to)i(the)f(study)h(of)f (the)h FB(p)p Fs(-adic)0 2257 y(v)-5 b(aluation)25 b(in)h(mo)r(dulus)g FB(p)h Fs(of)g(the)h(initial)23 b(terms)j(of)h(a)g(shifted)g(sum.)36 b(W)-7 b(e)27 b(no)n(w)g(fo)r(cus)g(on)g(harmonic)d(n)n(um)n(b)r(ers)i (of)h(the)0 2356 y(t)n(yp)r(e)h FB(H)7 b Fs(\()p FB(p)337 2326 y Fo(m)400 2356 y FB(n)p Fs(\).)0 2428 y SDict begin H.S end 0 2428 a 0 2428 a SDict begin 12 H.A end 0 2428 a 0 2428 a SDict begin [/View [/XYZ H.V]/Dest (subsection.3.1) cvn /DEST pdfmark end 0 2428 a 79 x Fs(3.1.)41 b FA(F)-8 b(orm)m(ulas)30 b(for)i FB(H)7 b Fs(\()p FB(p)880 2476 y Fo(m)943 2507 y FB(n)p Fs(\))p FA(.)41 b Fs(W)-7 b(e)28 b(recall)d(a)i(stronger)f(v)n(ersion)f(of)i(Lemma)2524 2507 y SDict begin H.S end 2524 2507 a 0 TeXcolorgray Fs(2.1)p 0 TeXcolorgray 2631 2453 a SDict begin H.R end 2631 2453 a 2631 2507 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.2.1) cvn H.B /ANN pdfmark end 2631 2507 a Fs(:)0 2689 y SDict begin H.S end 0 2689 a 0 2689 a SDict begin 12 H.A end 0 2689 a 0 2689 a SDict begin [/View [/XYZ H.V]/Dest (Theorem.3.5) cvn /DEST pdfmark end 0 2689 a 0 TeXcolorgray FA(Prop)s(osition)j(3.5)h(\(see)g Fs([)887 2689 y SDict begin H.S end 887 2689 a 0 TeXcolorgray Fs(3)p 0 TeXcolorgray 929 2636 a SDict begin H.R end 929 2636 a 929 2689 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Boyd) cvn H.B /ANN pdfmark end 929 2689 a Fs(])p FA(,)g(Lemma)f (3.1\).)p 0 TeXcolorgray 40 w Fh(If)e FB(p)23 b Fq(\025)f Fs(5)27 b Fh(is)g(prime)e(and)j FB(n)23 b Fq(\025)f Fs(1)p Fh(,)28 b(w)n(e)f(ha)n(v)n(e)1437 2872 y FB(H)7 b Fs(\()p FB(pn)p Fs(\))23 b(=)1790 2816 y FB(H)7 b Fs(\()p FB(n)p Fs(\))p 1790 2853 191 4 v 1864 2929 a FB(p)2008 2872 y Fs(+)18 b FB(O)r Fs(\()p FB(p)2230 2838 y Fp(2)2268 2872 y Fs(\))p FB(:)0 3066 y Fs(W)-7 b(e)28 b(could)e(conclude)h(that,) h(if)f FB(\027)1026 3078 y Fo(p)1064 3066 y Fs(\()p FB(H)7 b Fs(\()p FB(n)p Fs(\)\))24 b Fq(\024)f Fs(2,)k(then)1387 3200 y FB(\027)1428 3212 y Fo(p)1466 3200 y Fs(\()p FB(H)7 b Fs(\()p FB(pn)p Fs(\)\))24 b(=)f FB(\027)1915 3212 y Fo(p)1953 3200 y Fs(\()p FB(H)7 b Fs(\()p FB(n)p Fs(\)\))19 b Fq(\000)f Fs(1)p FB(:)0 3383 y SDict begin H.S end 0 3383 a 0 3383 a SDict begin 12 H.A end 0 3383 a 0 3383 a SDict begin [/View [/XYZ H.V]/Dest (Theorem.3.6) cvn /DEST pdfmark end 0 3383 a 0 TeXcolorgray FA(Lemma)30 b(3.6.)p 0 TeXcolorgray 40 w Fh(If)e FB(p)22 b Fq(\025)h Fs(5)k Fh(is)g(prime)e(and)j FB(n)23 b Fq(\025)f Fs(1)p Fh(,)27 b(w)n(e)h(ha)n(v)n(e)1437 3561 y FB(H)7 b Fs(\()p FB(p)1587 3527 y Fp(2)1624 3561 y FB(n)p Fs(\))23 b(=)1827 3505 y FB(H)7 b Fs(\()p FB(n)p Fs(\))p 1827 3542 191 4 v 1882 3618 a FB(p)1924 3594 y Fp(2)2045 3561 y Fs(+)19 b FB(O)r Fs(\()p FB(p)p Fs(\))p FB(;)p 0 TeXcolorgray 0 3755 a Fg(Pr)l(o)l(of.)p 0 TeXcolorgray 43 w Fs(Prop)r(osition)707 3755 y SDict begin H.S end 707 3755 a 0 TeXcolorgray Fs(3.5)p 0 TeXcolorgray 814 3701 a SDict begin H.R end 814 3701 a 814 3755 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.3.5) cvn H.B /ANN pdfmark end 814 3755 a 27 w Fs(implies)24 b(that)k(there)f(exists)f FB(\013)e Fq(2)f Fi(Q)39 b Fs(with)27 b FB(\027)2221 3767 y Fo(p)2260 3755 y Fs(\()p FB(\013)p Fs(\))d Fq(\025)e Fs(0)28 b(suc)n(h)f(that)1437 3941 y FB(H)7 b Fs(\()p FB(pn)p Fs(\))23 b(=)1790 3885 y FB(H)7 b Fs(\()p FB(n)p Fs(\))p 1790 3922 191 4 v 1864 3998 a FB(p)2008 3941 y Fs(+)18 b FB(O)r Fs(\()p FB(p)2230 3906 y Fp(2)2268 3941 y Fs(\))p FB(;)0 4115 y Fs(then)1119 4242 y FB(H)7 b Fs(\()p FB(p)1269 4208 y Fp(2)1306 4242 y FB(n)p Fs(\))23 b(=)1509 4186 y FB(H)7 b Fs(\()p FB(pn)p Fs(\))p 1509 4223 232 4 v 1604 4299 a FB(p)1769 4242 y Fs(+)18 b FB(O)r Fs(\()p FB(p)p Fs(\))24 b(=)2145 4186 y FB(H)7 b Fs(\()p FB(n)p Fs(\))p 2145 4223 191 4 v 2200 4299 a FB(p)2242 4275 y Fp(2)2363 4242 y Fs(+)19 b FB(O)r Fs(\()p FB(p)p Fs(\))p FB(:)3696 4398 y Ff(\003)0 4580 y SDict begin H.S end 0 4580 a 0 4580 a SDict begin 12 H.A end 0 4580 a 0 4580 a SDict begin [/View [/XYZ H.V]/Dest (Theorem.3.7) cvn /DEST pdfmark end 0 4580 a 0 TeXcolorgray FA(Lemma)30 b(3.7.)p 0 TeXcolorgray 40 w Fh(Let)d FB(a;)14 b(b)23 b Fq(2)g Fi(Z)p Fh(,)e(then)28 b Fs(\()p FB(b;)14 b(p)p Fs(\))23 b(=)g(1)k Fh(and)h FB(k)d Fq(\025)e Fs(0)k Fh(an)h(in)n(teger.)35 b(Then)1560 4712 y FB(a)p 1523 4749 119 4 v 1523 4825 a(p)1565 4801 y Fo(k)1606 4825 y FB(b)1675 4768 y Fs(=)1772 4712 y FB(ab)1852 4682 y Fk(\000)p Fp(1)p 1772 4749 169 4 v 1815 4825 a FB(p)1857 4801 y Fo(k)1969 4768 y Fs(+)18 b FB(O)r Fs(\()p FB(p)p Fs(\))p FB(;)0 4956 y Fh(where)27 b FB(b)276 4926 y Fk(\000)p Fp(1)393 4956 y Fh(denotes)g(the)h(in)n(v)n(erse)d(of)i FB(b)h Fh(mo)r(dulo)d FB(p)1608 4926 y Fo(k)q Fp(+1)1733 4956 y Fh(.)p 0 TeXcolorgray 0 5105 a Fg(Pr)l(o)l(of.)p 0 TeXcolorgray 43 w Fs(W)-7 b(e)28 b(ha)n(v)n(e)1092 5254 y FB(\027)1133 5266 y Fo(p)1185 5137 y Fl(\022)1293 5198 y FB(a)p 1256 5235 119 4 v 1256 5311 a(p)1298 5287 y Fo(k)1339 5311 y FB(b)1403 5254 y Fq(\000)1496 5198 y FB(ab)1576 5168 y Fk(\000)p Fp(1)p 1496 5235 169 4 v 1539 5311 a FB(p)1581 5287 y Fo(k)1674 5137 y Fl(\023)1759 5254 y Fs(=)22 b FB(\027)1887 5266 y Fo(p)1939 5137 y Fl(\022)2011 5198 y FB(a)p Fs(\(1)c Fq(\000)g FB(bb)2302 5168 y Fk(\000)p Fp(1)2390 5198 y Fs(\))p 2011 5235 412 4 v 2157 5311 a FB(p)2199 5287 y Fo(k)2240 5311 y FB(b)2432 5137 y Fl(\023)2517 5254 y Fq(\025)k Fs(1)p FB(:)3696 5415 y Ff(\003)0 5598 y SDict begin H.S end 0 5598 a 0 5598 a SDict begin 12 H.A end 0 5598 a 0 5598 a SDict begin [/View [/XYZ H.V]/Dest (Theorem.3.8) cvn /DEST pdfmark end 0 5598 a 0 TeXcolorgray FA(Lemma)30 b(3.8.)p 0 TeXcolorgray 40 w Fh(If)e FB(a;)14 b(b)22 b Fq(2)h Fi(Z)e Fh(and)28 b FB(a)23 b Fs(=)g FB(b)18 b Fs(+)g FB(O)r Fs(\()p FB(p)1505 5568 y Fo(k)q Fp(+1)1630 5598 y Fs(\))p Fh(,)1603 5720 y FB(a)p 1584 5757 83 4 v 1584 5833 a(p)1626 5809 y Fo(k)1700 5776 y Fs(=)1821 5720 y FB(b)p 1797 5757 V 1797 5833 a(p)1839 5809 y Fo(k)1908 5776 y Fs(+)g FB(O)r Fs(\()p FB(p)p Fs(\))p FB(:)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 5 5 TeXDict begin HPSdict begin 5 4 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -8 -191 a SDict begin H.S end -8 -191 a -8 -191 a SDict begin H.R end -8 -191 a -8 -191 a SDict begin [/View [/XYZ H.V]/Dest (page.5) cvn /DEST pdfmark end -8 -191 a 0 TeXcolorgray 155 -116 a Fo(p)p Fp(-ADIC)30 b(V)-7 b(ALUA)i(TION)29 b(OF)g(HARMONIC)h(SUMS)e(AND)h(THEIR)g (CONNECTIONS)h(WITH)f(W)n(OLSTENHOLME)f(PRIMES)122 b(5)p 0 TeXcolorgray 0 TeXcolorgray 0 83 a Fg(Pr)l(o)l(of.)p 0 TeXcolorgray 43 w Fs(W)-7 b(e)28 b(ha)n(v)n(e)977 230 y FB(\027)1018 242 y Fo(p)1070 113 y Fl(\022)1160 174 y FB(a)p 1141 211 83 4 v 1141 287 a(p)1183 263 y Fo(k)1252 230 y Fq(\000)1368 174 y FB(b)p 1345 211 V 1345 287 a(p)1387 263 y Fo(k)1438 113 y Fl(\023)1522 230 y Fs(=)22 b FB(\027)1650 242 y Fo(p)1703 113 y Fl(\022)1774 174 y FB(a)c Fq(\000)g FB(b)p 1774 211 181 4 v 1823 287 a(p)1865 263 y Fo(k)1965 113 y Fl(\023)2049 230 y Fs(=)k FB(\027)2177 242 y Fo(p)2216 230 y Fs(\()p FB(a)c Fq(\000)g FB(b)p Fs(\))h Fq(\000)f FB(k)26 b Fq(\025)c Fs(1)p FB(:)3696 393 y Ff(\003)0 546 y Fs(W)-7 b(e)28 b(no)n(w)f(recall)e(the)j(famous)e(Clausen-V)-7 b(an)25 b(Staudt)k(Theorem:)0 728 y SDict begin H.S end 0 728 a 0 728 a SDict begin 12 H.A end 0 728 a 0 728 a SDict begin [/View [/XYZ H.V]/Dest (Theorem.3.9) cvn /DEST pdfmark end 0 728 a 0 TeXcolorgray FA(Theorem)h(3.9)h(\(Clausen-V)-8 b(an)32 b(Staudt\).)p 0 TeXcolorgray 42 w Fh(Let)c FB(n)23 b Fq(2)g Fi(N)38 b Fh(and)27 b FB(B)2150 740 y Fo(i)2205 728 y Fh(b)r(e)h(the)g FB(i)p Fh(-th)g(Bernoulli)c(n)n(um)n(b)r(er,)i (then)1603 904 y FB(B)1666 916 y Fp(2)p Fo(n)1762 904 y Fs(+)1890 825 y Fl(X)1841 1003 y Fo(p)19 b Fp(prime)1818 1066 y(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(j)p Fp(2)p Fo(n)2106 848 y Fs(1)p 2106 885 42 4 v 2106 961 a FB(p)0 1190 y Fh(is)26 b(an)i(in)n(teger.)100 1339 y Fs(See)f([)270 1339 y SDict begin H.S end 270 1339 a 0 TeXcolorgray Fs(4)p 0 TeXcolorgray 312 1285 a SDict begin H.R end 312 1285 a 312 1339 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Carlitz) cvn H.B /ANN pdfmark end 312 1339 a Fs(])h(for)f(a)g(pro)r(of)g (of)g(the)h(theorem.)0 1522 y SDict begin H.S end 0 1522 a 0 1522 a SDict begin 12 H.A end 0 1522 a 0 1522 a SDict begin [/View [/XYZ H.V]/Dest (Theorem.3.10) cvn /DEST pdfmark end 0 1522 a 0 TeXcolorgray FA(Corollary)k(3.10.)p 0 TeXcolorgray 40 w Fh(Let)27 b FB(n)c Fq(2)h Fi(N)37 b Fh(and)28 b FB(B)1275 1534 y Fo(i)1330 1522 y Fh(b)r(e)g(the)g FB(i)p Fh(-th)f(Bernoulli)d(n)n(um)n(b)r(er,)i(then)p 0 TeXcolorgray 192 1640 a(i\))p 0 TeXcolorgray 41 w FB(\027)330 1652 y Fo(p)368 1640 y Fs(\()p FB(B)463 1652 y Fp(2)p Fo(n)542 1640 y Fs(\))d Fq(\025)g(\000)p Fs(1)p FB(;)p 0 TeXcolorgray 169 1739 a Fh(ii\))p 0 TeXcolorgray 40 w FB(\027)330 1751 y Fo(p)368 1739 y Fs(\()p FB(B)463 1751 y Fp(2)p Fo(n)542 1739 y Fs(\))g(=)g Fq(\000)p Fs(1)128 b Fq(\()-14 b(\))129 b FB(p)18 b Fq(\000)g Fs(1)23 b Fq(j)g Fs(2)p FB(n:)100 1889 y Fs(The)k(previous)f(Prop)r(ositions)d (and)28 b(Lemmas)d(are)i(used)g(to)h(in)n(v)n(estigate)c FB(H)7 b Fs(\()p FB(p)2564 1859 y Fo(m)2627 1889 y FB(n)p Fs(\))28 b(more)d(in)i(detail:)0 2072 y SDict begin H.S end 0 2072 a 0 2072 a SDict begin 12 H.A end 0 2072 a 0 2072 a SDict begin [/View [/XYZ H.V]/Dest (Theorem.3.11) cvn /DEST pdfmark end 0 2072 a 0 TeXcolorgray FA(Theorem)36 b(3.11.)p 0 TeXcolorgray 40 w Fh(Let)d FB(n;)14 b(m)32 b Fh(b)r(e)i(non-negativ)n(e)c(in)n(tegers)g(and)j FB(p)e Fq(\025)g Fs(5)i Fh(b)r(e)g(a)f(prime)f(n)n(um)n(b)r(er,)h(then)i(the)f (follo)n(wing)0 2171 y(form)n(ula)25 b(holds)481 2369 y SDict begin H.S end 481 2369 a 481 2369 a SDict begin 12 H.A end 481 2369 a 481 2369 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.1) cvn /DEST pdfmark end 481 2369 a FB(H)7 b Fs(\()p FB(p)631 2335 y Fo(m)694 2369 y FB(n)p Fs(\))23 b(=)896 2313 y FB(H)7 b Fs(\()p FB(n)p Fs(\))p 896 2350 191 4 v 939 2426 a FB(p)981 2402 y Fo(m)1115 2369 y Fs(+)1198 2265 y Fo(m)p Fk(\000)p Fp(1)1210 2290 y Fl(X)1209 2469 y Fo(h)p Fp(=2)1397 2265 y Fo(h)1356 2290 y Fl(X)1356 2469 y Fo(k)q Fp(=1)1500 2305 y FB(B)1563 2322 y Fo(p)1597 2305 y Fm(h)1636 2322 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(2)p Fo(k)p 1500 2350 433 4 v 1631 2426 a Fs(2)p FB(k)s(p)1761 2402 y Fo(h)1943 2252 y Fl(\022)2004 2313 y FB(p)2046 2283 y Fo(h)2089 2313 y Fs(\()p FB(p)18 b Fq(\000)g Fs(1\))h Fq(\000)f Fs(1)2127 2426 y(2)p FB(k)j Fq(\000)d Fs(1)2481 2252 y Fl(\023)2542 2369 y FB(p)2584 2335 y Fp(2)p Fo(k)q Fp(\()p Fo(m)p Fk(\000)p Fo(h)p Fp(\))2860 2369 y FB(n)2910 2335 y Fp(2)p Fo(k)3002 2369 y Fs(+)g FB(O)r Fs(\()p FB(p)p Fs(\))p FB(;)376 b Fs(\(1\))0 2580 y Fh(where)27 b FB(B)303 2592 y Fo(i)358 2580 y Fh(is)g(the)h FB(i)p Fh(-th)f(Bernoulli)d(n)n(um)n(b)r(er.)p 0 TeXcolorgray 0 2730 a Fg(Pr)l(o)l(of.)p 0 TeXcolorgray 43 w Fs(In)k(order)e(to)i (pro)n(v)n(e)e(the)i(statemen)n(t:)0 2829 y(Let)g(us)f(de\014ne)799 2992 y FB(a)843 3004 y Fo(m)906 2992 y Fs(\()p FB(n)p Fs(\))c(:=)1154 2888 y Fo(m)p Fk(\000)p Fp(1)1166 2913 y Fl(X)1164 3092 y Fo(h)p Fp(=2)1352 2888 y Fo(h)1312 2913 y Fl(X)1311 3092 y Fo(k)q Fp(=1)1456 2928 y FB(B)1519 2945 y Fo(p)1553 2929 y Fm(h)1592 2945 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(2)p Fo(k)p 1456 2973 V 1586 3049 a Fs(2)p FB(k)s(p)1716 3025 y Fo(h)1899 2875 y Fl(\022)1960 2936 y FB(p)2002 2906 y Fo(h)2044 2936 y Fs(\()p FB(p)c Fq(\000)f Fs(1\))g Fq(\000)g Fs(1)2083 3049 y(2)p FB(k)j Fq(\000)d Fs(1)2437 2875 y Fl(\023)2498 2992 y FB(p)2540 2958 y Fp(2)p Fo(k)q Fp(\()p Fo(m)p Fk(\000)p Fo(h)p Fp(\))2815 2992 y FB(n)2865 2958 y Fp(2)p Fo(k)2939 2992 y FB(:)0 3186 y Fs(Hence)28 b(the)g(statemen)n(t)e(b)r (ecomes:)1263 3350 y FB(H)7 b Fs(\()p FB(p)1413 3315 y Fo(m)1476 3350 y FB(n)p Fs(\))23 b(=)1678 3294 y FB(H)7 b Fs(\()p FB(n)p Fs(\))p 1678 3331 191 4 v 1721 3407 a FB(p)1763 3383 y Fo(m)1897 3350 y Fs(+)18 b FB(a)2024 3362 y Fo(m)2087 3350 y Fs(\()p FB(n)p Fs(\))h(+)f FB(O)r Fs(\()p FB(p)p Fs(\))p FB(:)0 3530 y Fs(The)25 b(pro)r(of)g(follo)n(ws) d(b)n(y)j(induction)f(on)h FB(m)p Fs(:)36 b(our)24 b(aim)f(is)i(to)g (\014nd)g(functions)g FB(\037)2434 3542 y Fo(m)2497 3530 y Fs(\()p FB(n)p Fs(\))h(suc)n(h)f(that)g(the)h(follo)n(wing)21 b(congru-)0 3630 y(ences)27 b(hold:)75 3794 y FB(H)7 b Fs(\()p FB(p)225 3760 y Fo(m)p Fp(+1)372 3794 y FB(n)p Fs(\))23 b(=)575 3738 y FB(H)7 b Fs(\()p FB(pn)p Fs(\))p 575 3775 232 4 v 639 3851 a FB(p)681 3827 y Fo(m)835 3794 y Fs(+)18 b FB(a)962 3806 y Fo(m)1025 3794 y Fs(\()p FB(n)p Fs(\))h(+)f FB(O)r Fs(\()p FB(p)p Fs(\))24 b(=)1534 3738 y FB(H)7 b Fs(\()p FB(n)p Fs(\))p 1534 3775 191 4 v 1535 3851 a FB(p)1577 3827 y Fo(m)p Fp(+1)1752 3794 y Fs(+)18 b FB(\037)1887 3806 y Fo(m)1950 3794 y Fs(\()p FB(n)p Fs(\))h(+)f FB(a)2210 3806 y Fo(m)2273 3794 y Fs(\()p FB(n)p Fs(\))h(+)f FB(O)r Fs(\()p FB(p)p Fs(\))24 b(=)2782 3738 y FB(H)7 b Fs(\()p FB(n)p Fs(\))p 2782 3775 V 2783 3851 a FB(p)2825 3827 y Fo(m)p Fp(+1)3000 3794 y Fs(+)18 b FB(a)3127 3806 y Fo(m)p Fp(+1)3274 3794 y Fs(\()p FB(n)p Fs(\))h(+)f FB(O)r Fs(\()p FB(p)p Fs(\))p FB(:)p 0 TeXcolorgray 0 3975 a Fg(Step)30 b(1.)p 0 TeXcolorgray 42 w Fs(Find)d FB(\037)543 3987 y Fo(m)606 3975 y Fs(\()p FB(n)p Fs(\))h(suc)n(h)g(that)1290 4068 y FB(H)7 b Fs(\()p FB(pn)p Fs(\))p 1290 4105 232 4 v 1354 4181 a FB(p)1396 4157 y Fo(m)1555 4124 y Fs(=)1653 4068 y FB(H)g Fs(\()p FB(n)p Fs(\))p 1653 4105 191 4 v 1654 4181 a FB(p)1696 4157 y Fo(m)p Fp(+1)1871 4124 y Fs(+)18 b FB(\037)2006 4136 y Fo(m)2069 4124 y Fs(\()p FB(n)p Fs(\))h(+)f FB(O)r Fs(\()p FB(p)p Fs(\))p FB(:)p 0 TeXcolorgray 0 4288 a Fg(Step)30 b(2.)p 0 TeXcolorgray 42 w Fs(Pro)n(v)n(e)25 b(that,)j(for)f(the)h FB(\037)1053 4300 y Fo(m)1116 4288 y Fs(\()p FB(n)p Fs(\))g(already)d(found,)j(it)g(holds)1227 4424 y FB(\037)1279 4436 y Fo(m)1342 4424 y Fs(\()p FB(pn)p Fs(\))19 b(+)f FB(a)1644 4436 y Fo(m)1707 4424 y Fs(\()p FB(n)p Fs(\))23 b(=)g FB(a)1976 4436 y Fo(m)p Fp(+1)2123 4424 y Fs(\()p FB(n)p Fs(\))c(+)f FB(O)r Fs(\()p FB(p)p Fs(\))p FB(:)0 4561 y Fg(Pr)l(o)l(of)31 b(of)f(step)g(1.)38 b Fs(By)27 b(Prop)r(osition)1174 4561 y SDict begin H.S end 1174 4561 a 0 TeXcolorgray Fs(3.5)p 0 TeXcolorgray 1281 4508 a SDict begin H.R end 1281 4508 a 1281 4561 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.3.5) cvn H.B /ANN pdfmark end 1281 4561 a 27 w Fs(and)h(noting)e(that)1267 4776 y FB(H)7 b Fs(\()p FB(pn)p Fs(\))23 b(=)1649 4672 y Fo(n)1610 4697 y Fl(X)1609 4876 y Fo(k)q Fp(=1)1777 4720 y Fs(1)p 1754 4757 88 4 v 1754 4833 a FB(pk)1870 4776 y Fs(+)1953 4672 y Fo(n)p Fk(\000)p Fp(1)1956 4697 y Fl(X)1960 4876 y Fo(`)p Fp(=0)2094 4668 y Fo(p)p Fk(\000)p Fp(1)2093 4697 y Fl(X)2093 4876 y Fo(k)q Fp(=1)2328 4720 y Fs(1)p 2237 4757 224 4 v 2237 4833 a FB(`p)18 b Fs(+)g FB(k)2471 4776 y(;)0 4987 y Fs(w)n(e)27 b(obtain)1006 5095 y FB(H)7 b Fs(\()p FB(pn)p Fs(\))p 1006 5132 232 4 v 1070 5208 a FB(p)1112 5184 y Fo(m)1271 5151 y Fs(=)1401 5095 y(1)p 1369 5132 105 4 v 1369 5208 a FB(p)1411 5184 y Fo(m)1498 5009 y Fl( )1603 5047 y Fo(n)1564 5072 y Fl(X)1563 5251 y Fo(k)q Fp(=1)1731 5095 y Fs(1)p 1708 5132 88 4 v 1708 5208 a FB(pk)1824 5151 y Fs(+)1907 5047 y Fo(n)p Fk(\000)p Fp(1)1910 5072 y Fl(X)1914 5251 y Fo(`)p Fp(=0)2048 5043 y Fo(p)p Fk(\000)p Fp(1)2047 5072 y Fl(X)2047 5251 y Fo(k)q Fp(=1)2282 5095 y Fs(1)p 2191 5132 224 4 v 2191 5208 a FB(`p)18 b Fs(+)g FB(k)2425 5009 y Fl(!)2509 5151 y Fs(+)g FB(O)r Fs(\()p FB(p)p Fs(\))1271 5446 y(=)1401 5390 y(1)p 1369 5427 105 4 v 1369 5503 a FB(p)1411 5479 y Fo(m)1498 5304 y Fl( )1573 5390 y FB(H)7 b Fs(\()p FB(n)p Fs(\))p 1573 5427 191 4 v 1648 5503 a FB(p)1792 5446 y Fs(+)1875 5342 y Fo(n)p Fk(\000)p Fp(1)1878 5367 y Fl(X)1882 5546 y Fo(`)p Fp(=0)2016 5338 y Fo(p)p Fk(\000)p Fp(1)2015 5367 y Fl(X)2015 5546 y Fo(k)q Fp(=1)2250 5390 y Fs(1)p 2159 5427 224 4 v 2159 5503 a FB(`p)18 b Fs(+)g FB(k)2393 5304 y Fl(!)2477 5446 y Fs(+)g FB(O)r Fs(\()p FB(p)p Fs(\))1271 5741 y(=)1369 5685 y FB(H)7 b Fs(\()p FB(n)p Fs(\))p 1369 5722 191 4 v 1370 5798 a FB(p)1412 5774 y Fo(m)p Fp(+1)1588 5741 y Fs(+)1712 5685 y(1)p 1681 5722 105 4 v 1681 5798 a FB(p)1723 5774 y Fo(m)1809 5637 y(n)p Fk(\000)p Fp(1)1812 5662 y Fl(X)1816 5841 y Fo(`)p Fp(=0)1950 5633 y Fo(p)p Fk(\000)p Fp(1)1949 5662 y Fl(X)1949 5841 y Fo(k)q Fp(=1)2185 5685 y Fs(1)p 2094 5722 224 4 v 2094 5798 a FB(`p)17 b Fs(+)h FB(k)2346 5741 y Fs(+)g FB(O)r Fs(\()p FB(p)p Fs(\))p FB(:)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 6 6 TeXDict begin HPSdict begin 6 5 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -8 -191 a SDict begin H.S end -8 -191 a -8 -191 a SDict begin H.R end -8 -191 a -8 -191 a SDict begin [/View [/XYZ H.V]/Dest (page.6) cvn /DEST pdfmark end -8 -191 a 0 TeXcolorgray 0 -116 a Fp(6)733 b(LEONARDO)23 b(CAR)n(OFIGLIO,)g(LUIGI)f(DE)g(FILPO,)g(ALESSANDR)n(O)g(GAMBINI)p 0 TeXcolorgray 0 83 a Fs(Hence,)28 b(w)n(e)f(found)1370 268 y FB(\037)1422 280 y Fo(m)1485 268 y Fs(\()p FB(n)p Fs(\))d(=)1752 212 y(1)p 1720 249 105 4 v 1720 325 a FB(p)1762 301 y Fo(m)1849 165 y(n)p Fk(\000)p Fp(1)1852 189 y Fl(X)1856 368 y Fo(`)p Fp(=0)1990 161 y Fo(p)p Fk(\000)p Fp(1)1989 189 y Fl(X)1989 368 y Fo(k)q Fp(=1)2225 212 y Fs(1)p 2134 249 224 4 v 2134 325 a FB(`p)17 b Fs(+)h FB(k)2367 268 y(:)0 467 y Fs(W)-7 b(e)28 b(fo)r(cus)g(on)f(the)h(inner) e(sum)h(of)h(the)g(RHS:)699 615 y(1)p 667 652 105 4 v 667 728 a FB(p)709 704 y Fo(m)796 564 y(p)p Fk(\000)p Fp(1)796 592 y Fl(X)796 771 y Fo(k)q Fp(=1)1031 615 y Fs(1)p 940 652 224 4 v 940 728 a FB(`p)18 b Fs(+)g FB(k)1197 671 y Fs(=)1326 615 y(1)p 1294 652 105 4 v 1294 728 a FB(p)1336 704 y Fo(m)1424 564 y(p)p Fk(\000)p Fp(1)1424 592 y Fl(X)1423 771 y Fo(k)q Fp(=1)1544 671 y Fs(\()p FB(`p)g Fs(+)g FB(k)s Fs(\))1832 637 y Fo(p)1866 612 y Fm(m)1921 637 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(1)2200 671 y Fs(+)g FB(O)r Fs(\()p FB(p)p Fs(\))1197 969 y(=)1326 913 y(1)p 1294 950 V 1294 1026 a FB(p)1336 1002 y Fo(m)1423 827 y Fl( )1489 861 y Fo(`p)p Fp(+)p Fo(p)p Fk(\000)p Fp(1)1545 890 y Fl(X)1545 1069 y Fo(k)q Fp(=1)1735 969 y FB(k)1781 935 y Fo(p)1815 910 y Fm(m)1870 935 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(1)2149 969 y Fq(\000)2261 861 y Fo(`p)2232 890 y Fl(X)2232 1069 y Fo(k)q Fp(=1)2366 969 y FB(k)2412 935 y Fo(p)2446 910 y Fm(m)2502 935 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(1)2762 827 y Fl(!)2846 969 y Fs(+)g FB(O)r Fs(\()p FB(p)p Fs(\))1197 1281 y(=)1326 1224 y(1)p 1294 1261 V 1294 1337 a FB(p)1336 1313 y Fo(m)1423 1114 y Fl(0)1423 1263 y(@)1496 1170 y Fp(\()p Fo(`)p Fp(+1\))p Fo(p)1535 1202 y Fl(X)1534 1380 y Fo(k)q Fp(=1)1708 1281 y FB(k)1754 1246 y Fo(p)1788 1221 y Fm(m)1843 1246 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(1)2121 1281 y Fq(\000)2234 1173 y Fo(`p)2205 1202 y Fl(X)2204 1380 y Fo(k)q Fp(=1)2339 1281 y FB(k)2385 1246 y Fo(p)2419 1221 y Fm(m)2474 1246 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(1)2734 1114 y Fl(1)2734 1263 y(A)2826 1281 y Fs(+)g FB(O)r Fs(\()p FB(p)p Fs(\))p FB(:)3655 1281 y SDict begin H.S end 3655 1281 a 3655 1281 a SDict begin 15 H.A end 3655 1281 a 3655 1281 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.2) cvn /DEST pdfmark end 3655 1281 a Fs(\(2\))0 1533 y(W)-7 b(e)38 b(ha)n(v)n(e)e(tak)n(en)g(adv)-5 b(an)n(tage)36 b(of)h(the)h(fact)f(that,)j(if)d FB(y)j Fs(has)c(an)h(in)n(v)n(erse)e (mo)r(dulo)g FB(n)p Fs(,)40 b FB(y)2834 1503 y Fk(\000)p Fp(1)2961 1533 y Fq(\021)f FB(y)3109 1503 y Fo(')p Fp(\()p Fo(n)p Fp(\))p Fk(\000)p Fp(1)3335 1533 y Fs(\(mo)r(d)27 b FB(n)p Fs(\))37 b(b)n(y)0 1633 y(Euler's)c(Theorem,)h(where)g FB(')h Fs(denotes)f(Euler's)f(totien)n(t)g(function.)57 b(This)34 b(explains)e(the)j(c)n(hoice)d(of)j(the)g(exp)r(onen)n(t)0 1733 y FB(p)42 1703 y Fo(m)105 1733 y Fs(\()p FB(p)18 b Fq(\000)g Fs(1\))23 b(=)g FB(')p Fs(\()p FB(p)593 1703 y Fo(m)p Fp(+1)740 1733 y Fs(\).)0 1832 y(No)n(w)k(the)h(recurring)d (expression)1588 1964 y(1)p 1556 2001 105 4 v 1556 2077 a FB(p)1598 2053 y Fo(m)1714 1913 y(`p)1685 1942 y Fl(X)1685 2120 y Fo(k)q Fp(=1)1819 2021 y FB(k)1865 1986 y Fo(p)1899 1961 y Fm(m)1954 1986 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(1)0 2224 y Fs(can)d(b)r(e)h(simpli\014ed)c(using)i (Bernoulli's)e(form)n(ula)h(for)i(sums)f(of)i(p)r(o)n(w)n(ers)e(\(F)-7 b(aulhab)r(er's)21 b(form)n(ula,)g(see)h([)3181 2224 y SDict begin H.S end 3181 2224 a 0 TeXcolorgray Fs(5)p 0 TeXcolorgray 3223 2171 a SDict begin H.R end 3223 2171 a 3223 2224 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Conway) cvn H.B /ANN pdfmark end 3223 2224 a -1 w Fs(,)i Fj(\247)p Fs(4])e(pag.)35 b(106\):)598 2402 y(1)p 567 2439 105 4 v 567 2515 a FB(p)609 2491 y Fo(m)725 2350 y(`p)696 2379 y Fl(X)695 2560 y Fo(k)q Fp(=1)830 2458 y FB(k)876 2424 y Fo(p)910 2399 y Fm(m)965 2424 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(1)1248 2458 y Fs(=)1519 2402 y(1)p 1346 2439 388 4 v 1346 2515 a FB(p)1388 2491 y Fp(2)p Fo(m)1484 2515 y Fs(\()p FB(p)18 b Fq(\000)g Fs(1\))1646 2347 y Fo(p)1680 2322 y Fm(m)1736 2347 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(1)1759 2379 y Fl(X)1759 2558 y Fo(k)q Fp(=0)1895 2458 y Fs(\()p Fq(\000)p Fs(1\))2066 2424 y Fo(k)2106 2458 y FB(B)2169 2470 y Fo(k)2210 2341 y Fl(\022)2271 2402 y FB(p)2313 2372 y Fo(m)2376 2402 y Fs(\()p FB(p)h Fq(\000)f Fs(1\))2426 2515 y FB(k)2626 2341 y Fl(\023)2687 2458 y Fs(\()p FB(`p)p Fs(\))2828 2424 y Fo(p)2862 2399 y Fm(m)2917 2424 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fo(k)3181 2458 y FB(:)3655 2458 y SDict begin H.S end 3655 2458 a 3655 2458 a SDict begin 15 H.A end 3655 2458 a 3655 2458 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.3) cvn /DEST pdfmark end 3655 2458 a Fs(\(3\))0 2680 y(In)24 b(terms)f(of)h (congruence)f(mo)r(dulo)f FB(p)p Fs(,)j(the)f(terms)f(with)h(0)e Fq(\024)h FB(k)j(<)c(p)2132 2650 y Fo(m)2195 2680 y Fs(\()p FB(p)12 b Fq(\000)g Fs(1\))g Fq(\000)g Fs(2)p FB(m)21 b Fs(v)-5 b(anish)23 b(since)g(they)h(ha)n(v)n(e)f(p)r(ositiv)n(e)0 2780 y FB(p)p Fs(-adic)35 b(v)-5 b(aluation:)54 b(b)n(y)37 b(Corollary)1169 2780 y SDict begin H.S end 1169 2780 a 0 TeXcolorgray Fs(3.10)p 0 TeXcolorgray 1318 2727 a SDict begin H.R end 1318 2727 a 1318 2780 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.3.10) cvn H.B /ANN pdfmark end 1318 2780 a 36 w Fs(w)n(e)g(\014nd)h(that)f FB(\027)1892 2792 y Fo(p)1931 2780 y Fs(\()p FB(B)2026 2792 y Fo(k)2067 2780 y Fs(\))i Fq(\025)g(\000)p Fs(1.)65 b(Moreo)n(v)n(er,)36 b(the)i FB(p)p Fs(-adic)d(v)-5 b(aluation)35 b(of)i(a)0 2883 y(binomial)19 b(co)r(e\016cien)n(t)j(is)h(non-negativ)n(e)d(and)k (b)n(y)f(remo)n(ving)d FB(B)1972 2895 y Fp(1)2032 2883 y Fs(w)n(e)j(do)g(not)h(need)f(the)h(factor)f(\()p Fq(\000)p Fs(1\))3138 2852 y Fo(k)3202 2883 y Fs(b)r(ecause)g FB(B)3568 2895 y Fo(k)3631 2883 y Fs(=)g(0)0 2982 y(for)k(all)e FB(k)h Fq(\025)d Fs(3)k(o)r(dd.)37 b(Then,)28 b(the)g(RHS)g(of)34 b(\()1377 2982 y SDict begin H.S end 1377 2982 a 0 TeXcolorgray Fs(3)p 0 TeXcolorgray 1419 2929 a SDict begin H.R end 1419 2929 a 1419 2982 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.3) cvn H.B /ANN pdfmark end 1419 2982 a Fs(\))28 b(is)e(equal)g(to)1237 3147 y(1)p 1064 3184 388 4 v 1064 3260 a FB(p)1106 3236 y Fp(2)p Fo(m)1202 3260 y Fs(\()p FB(p)18 b Fq(\000)g Fs(1\))1489 3099 y Fp(2)p Fo(m)1475 3124 y Fl(X)1475 3303 y Fo(k)q Fp(=1)1609 3203 y FB(B)1672 3218 y Fo(p)1706 3201 y Fm(m)1762 3218 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fo(k)2026 3086 y Fl(\022)2087 3147 y FB(p)2129 3117 y Fo(m)2192 3147 y Fs(\()p FB(p)g Fq(\000)g Fs(1\))2241 3260 y FB(k)2441 3086 y Fl(\023)2502 3203 y Fs(\()p FB(`p)p Fs(\))2643 3169 y Fo(k)2684 3203 y FB(:)0 3424 y Fs(Resuming)25 b(from)i(\()613 3424 y SDict begin H.S end 613 3424 a 0 TeXcolorgray Fs(2)p 0 TeXcolorgray 655 3371 a SDict begin H.R end 655 3371 a 655 3424 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.2) cvn H.B /ANN pdfmark end 655 3424 a Fs(\),)h(w)n(e)f(arriv)n(e)d(at)343 3609 y(1)p 311 3646 105 4 v 311 3722 a FB(p)353 3698 y Fo(m)441 3558 y(p)p Fk(\000)p Fp(1)440 3587 y Fl(X)440 3765 y Fo(k)q Fp(=1)676 3609 y Fs(1)p 585 3646 224 4 v 585 3722 a FB(`p)17 b Fs(+)h FB(k)841 3666 y Fs(=)971 3609 y(1)p 939 3646 105 4 v 939 3722 a FB(p)981 3698 y Fo(m)1068 3499 y Fl(0)1068 3648 y(@)1140 3555 y Fp(\()p Fo(`)p Fp(+1\))p Fo(p)1179 3587 y Fl(X)1179 3767 y Fo(k)q Fp(=1)1352 3666 y FB(k)1398 3631 y Fo(p)1432 3606 y Fm(m)1487 3631 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(1)1766 3666 y Fq(\000)1878 3558 y Fo(`p)1849 3587 y Fl(X)1849 3767 y Fo(k)q Fp(=1)1983 3666 y FB(k)2029 3631 y Fo(p)2063 3606 y Fm(m)2119 3631 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(1)2379 3499 y Fl(1)2379 3648 y(A)2470 3666 y Fs(+)g FB(O)r Fs(\()p FB(p)p Fs(\))841 3977 y(=)1112 3921 y(1)p 939 3958 388 4 v 939 4034 a FB(p)981 4010 y Fp(2)p Fo(m)1077 4034 y Fs(\()p FB(p)g Fq(\000)g Fs(1\))1364 3873 y Fp(2)p Fo(m)1350 3898 y Fl(X)1350 4077 y Fo(k)q Fp(=1)1485 3977 y FB(B)1548 3992 y Fo(p)1582 3975 y Fm(m)1637 3992 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fo(k)1901 3860 y Fl(\022)1962 3921 y FB(p)2004 3891 y Fo(m)2067 3921 y Fs(\()p FB(p)g Fq(\000)g Fs(1\))2116 4034 y FB(k)2316 3860 y Fl(\023)2391 3885 y(\020)2440 3977 y Fs(\()q(\()p FB(`)g Fs(+)g(1\))p FB(p)p Fs(\))2789 3935 y Fo(k)2848 3977 y Fq(\000)g Fs(\()p FB(`p)p Fs(\))3072 3943 y Fo(k)3113 3885 y Fl(\021)3181 3977 y Fs(+)g FB(O)r Fs(\()p FB(p)p Fs(\))p FB(:)0 4198 y Fs(No)n(w)23 b(w)n(e)g(ha)n(v)n(e)g(a)g (candidate)g FB(\037)983 4210 y Fo(m)1046 4198 y Fs(\()p FB(n)p Fs(\),)i(and)e(b)n(y)g(rearranging)d(terms,)j(dev)n(eloping)e (the)j(telescopic)d(sum)i(and)g(highligh)n(t-)0 4298 y(ing)j(again)g(that)i FB(B)603 4310 y Fo(k)667 4298 y Fs(=)22 b(0)27 b(for)g(all)f FB(k)g Fq(\025)c Fs(3)27 b(o)r(dd,)h(w)n(e)f(arriv)n(e)e(at)j(the)g(follo)n(wing)387 4597 y FB(\037)439 4609 y Fo(m)502 4597 y Fs(\()p FB(n)p Fs(\))23 b(=)910 4541 y(1)p 737 4578 V 737 4654 a FB(p)779 4630 y Fp(2)p Fo(m)875 4654 y Fs(\()p FB(p)18 b Fq(\000)g Fs(1\))1148 4493 y Fo(n)p Fk(\000)p Fp(1)1151 4518 y Fl(X)1155 4697 y Fo(`)p Fp(=0)1302 4493 y(2)p Fo(m)1288 4518 y Fl(X)1288 4697 y Fo(k)q Fp(=1)1422 4597 y FB(B)1485 4612 y Fo(p)1519 4596 y Fm(m)1575 4612 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fo(k)1838 4480 y Fl(\022)1900 4541 y FB(p)1942 4511 y Fo(m)2004 4541 y Fs(\()p FB(p)h Fq(\000)f Fs(1\))2054 4654 y FB(k)2254 4480 y Fl(\023)2329 4505 y(\020)2378 4597 y Fs(\()q(\()p FB(`)g Fs(+)g(1\))p FB(p)p Fs(\))2727 4556 y Fo(k)2786 4597 y Fq(\000)g Fs(\()p FB(`p)p Fs(\))3010 4563 y Fo(k)3051 4505 y Fl(\021)3119 4597 y Fs(+)g FB(O)r Fs(\()p FB(p)p Fs(\))639 4888 y(=)910 4832 y(1)p 737 4869 V 737 4945 a FB(p)779 4921 y Fp(2)p Fo(m)875 4945 y Fs(\()p FB(p)g Fq(\000)g Fs(1\))1162 4784 y Fp(2)p Fo(m)1148 4809 y Fl(X)1148 4988 y Fo(k)q Fp(=1)1282 4888 y FB(B)1345 4903 y Fo(p)1379 4887 y Fm(m)1435 4903 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fo(k)1699 4771 y Fl(\022)1760 4832 y FB(p)1802 4802 y Fo(m)1865 4832 y Fs(\()p FB(p)g Fq(\000)g Fs(1\))1914 4945 y FB(k)2114 4771 y Fl(\023)2175 4888 y Fs(\()p FB(np)p Fs(\))2331 4854 y Fo(k)2390 4888 y Fs(+)g FB(O)r Fs(\()p FB(p)p Fs(\))639 5179 y(=)768 5123 y(1)p 737 5160 105 4 v 737 5236 a FB(p)779 5212 y Fo(m)880 5075 y Fp(2)p Fo(m)866 5100 y Fl(X)865 5279 y Fo(k)q Fp(=1)1010 5117 y FB(B)1073 5132 y Fo(p)1107 5116 y Fm(m)1162 5132 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fo(k)p 1010 5160 417 4 v 1195 5236 a FB(k)1436 5062 y Fl(\022)1497 5123 y FB(p)1539 5093 y Fo(m)1602 5123 y Fs(\()p FB(p)h Fq(\000)f Fs(1\))g Fq(\000)g Fs(1)1651 5236 y FB(k)k Fq(\000)c Fs(1)1994 5062 y Fl(\023)2055 5179 y Fs(\()p FB(np)p Fs(\))2211 5145 y Fo(k)2271 5179 y Fs(+)g FB(O)r Fs(\()p FB(p)p Fs(\))639 5458 y(=)758 5354 y Fo(m)727 5379 y Fl(X)727 5557 y Fo(k)q Fp(=1)871 5396 y FB(B)934 5411 y Fo(p)968 5394 y Fm(m)1024 5411 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(2)p Fo(k)p 871 5438 450 4 v 1000 5514 a Fs(2)p FB(k)s(p)1130 5490 y Fo(m)1331 5340 y Fl(\022)1392 5401 y FB(p)1434 5371 y Fo(m)1497 5401 y Fs(\()p FB(p)g Fq(\000)g Fs(1\))g Fq(\000)g Fs(1)1525 5514 y(2)p FB(k)j Fq(\000)d Fs(1)1889 5340 y Fl(\023)1950 5458 y Fs(\()p FB(np)p Fs(\))2106 5423 y Fp(2)p Fo(k)2199 5458 y Fs(+)g FB(O)r Fs(\()p FB(p)p Fs(\))p FB(:)0 5674 y Fg(Pr)l(o)l(of)31 b(of)f(step)g(2.)38 b Fs(In)28 b(order)e(to)i(conclude)e(the)i(global)c (pro)r(of)j(w)n(e)h(need)f(to)h(sho)n(w)e(that)i(it)f(holds)1227 5816 y FB(a)1271 5828 y Fo(m)p Fp(+1)1418 5816 y Fs(\()p FB(n)p Fs(\))c(=)g FB(\037)1695 5828 y Fo(m)1758 5816 y Fs(\()p FB(pn)p Fs(\))c(+)f FB(a)2060 5828 y Fo(m)2123 5816 y Fs(\()p FB(n)p Fs(\))h(+)f FB(O)r Fs(\()p FB(p)p Fs(\))p FB(;)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 7 7 TeXDict begin HPSdict begin 7 6 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -8 -191 a SDict begin H.S end -8 -191 a -8 -191 a SDict begin H.R end -8 -191 a -8 -191 a SDict begin [/View [/XYZ H.V]/Dest (page.7) cvn /DEST pdfmark end -8 -191 a 0 TeXcolorgray 155 -116 a Fo(p)p Fp(-ADIC)30 b(V)-7 b(ALUA)i(TION)29 b(OF)g(HARMONIC)h(SUMS)e(AND)h(THEIR)g (CONNECTIONS)h(WITH)f(W)n(OLSTENHOLME)f(PRIMES)122 b(7)p 0 TeXcolorgray 0 83 a Fs(The)32 b(iden)n(tit)n(y)f(is)g(true)i(for)e FB(m)g Fs(=)g(3)h(and,)i(giv)n(en)c(the)j(inductiv)n(e)d(h)n(yp)r (othesis,)j(the)f(iden)n(tit)n(y)f(is)g(trivial)e(since)i FB(\037)3583 95 y Fo(m)3646 83 y Fs(\()p FB(n)p Fs(\))0 183 y(just)d(found)g(is)f(exactly)f(the)i FB(m)p Fs(-th)f(term)g(of)g (the)h(sum)f(of)h FB(a)1867 195 y Fo(m)1930 183 y Fs(\()p FB(n)p Fs(\).)1629 b Ff(\003)0 293 y SDict begin H.S end 0 293 a 0 293 a SDict begin 12 H.A end 0 293 a 0 293 a SDict begin [/View [/XYZ H.V]/Dest (section.4) cvn /DEST pdfmark end 0 293 a 1188 372 a Fs(4.)42 b Fr(Pr)n(oof)31 b(of)g(Theorem)2119 372 y SDict begin H.S end 2119 372 a 0 TeXcolorgray Fr(2.5)p 0 TeXcolorgray 2238 318 a SDict begin H.R end 2238 318 a 2238 372 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.2.5) cvn H.B /ANN pdfmark end 2238 372 a 31 w Fr(and)2454 372 y SDict begin H.S end 2454 372 a 0 TeXcolorgray Fr(2.6)p 0 TeXcolorgray 2573 318 a SDict begin H.R end 2573 318 a 2573 372 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.2.6) cvn H.B /ANN pdfmark end 2573 372 a 0 521 a Fs(W)-7 b(e)28 b(recall)d(here)i(this)g(famous)f (result)g(from)g(Kummer:)0 704 y SDict begin H.S end 0 704 a 0 704 a SDict begin 12 H.A end 0 704 a 0 704 a SDict begin [/View [/XYZ H.V]/Dest (Theorem.4.1) cvn /DEST pdfmark end 0 704 a 0 TeXcolorgray FA(Theorem)37 b(4.1)h(\(Kummer's)d (Congruence\).)p 0 TeXcolorgray 41 w Fh(If)f FB(a;)14 b(b)32 b Fq(2)h Fi(N)43 b Fh(are)33 b(ev)n(en,)h(not)g(divisible)29 b(b)n(y)34 b FB(p)22 b Fq(\000)g Fs(1)33 b Fh(and)g FB(a)g Fs(=)f FB(b)22 b Fs(+)0 804 y FB(O)r Fs(\()p FB(p)d Fq(\000)f Fs(1\))p Fh(,)28 b(then)1567 896 y FB(B)1630 908 y Fo(a)p 1567 933 104 4 v 1597 1010 a FB(a)1703 953 y Fs(=)1801 896 y FB(B)1864 908 y Fo(b)p 1801 933 97 4 v 1831 1010 a FB(b)1926 953 y Fs(+)18 b FB(O)r Fs(\()p FB(p)p Fs(\))p FB(:)0 1125 y Fs(No)n(w)27 b(w)n(e)g(are)g(able)f(to)i(pro)n(v)n(e)e (Theorem)1301 1125 y SDict begin H.S end 1301 1125 a 0 TeXcolorgray Fs(2.5)p 0 TeXcolorgray 1408 1072 a SDict begin H.R end 1408 1072 a 1408 1125 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.2.5) cvn H.B /ANN pdfmark end 1408 1125 a Fs(.)0 1205 y SDict begin H.S end 0 1205 a 0 1205 a SDict begin 12 H.A end 0 1205 a 0 1205 a SDict begin [/View [/XYZ H.V]/Dest (subsection.4.1) cvn /DEST pdfmark end 0 1205 a 84 x Fs(4.1.)41 b FA(Pro)s(of)31 b(of)h(Theorem)949 1289 y SDict begin H.S end 949 1289 a 0 TeXcolorgray FA(2.5)p 0 TeXcolorgray 1072 1235 a SDict begin H.R end 1072 1235 a 1072 1289 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.2.5) cvn H.B /ANN pdfmark end 1072 1289 a FA(.)p 0 TeXcolorgray 0 1448 a Fg(Pr)l(o)l(of)f(of)f(The)l(or)l(em)665 1448 y SDict begin H.S end 665 1448 a 0 TeXcolorgray Fg(2.5)p 0 TeXcolorgray 774 1395 a SDict begin H.R end 774 1395 a 774 1448 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.2.5) cvn H.B /ANN pdfmark end 774 1448 a Fg(.)p 0 TeXcolorgray 0 1485 a SDict begin H.S end 0 1485 a 0 1485 a SDict begin 12 H.A end 0 1485 a 0 1485 a SDict begin [/View [/XYZ H.V]/Dest (Item.7) cvn /DEST pdfmark end 0 1485 a 0 TeXcolorgray 141 1577 a Fs(\(1\))p 0 TeXcolorgray 42 w(If)g(w)n(e)g(pro)n(v)n(e)f(that)i FB(H)7 b Fs(\()p FB(p)1059 1547 y Fp(3)1096 1577 y FB(n)p Fs(\))27 b(=)g FB(O)r Fs(\()p FB(p)p Fs(\))32 b(it)d(follo)n(ws)f(from)h(Lemma)2359 1577 y SDict begin H.S end 2359 1577 a 0 TeXcolorgray Fs(3.6)p 0 TeXcolorgray 2466 1524 a SDict begin H.R end 2466 1524 a 2466 1577 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.3.6) cvn H.B /ANN pdfmark end 2466 1577 a 29 w Fs(that)2688 1537 y Fo(H)t Fp(\()p Fo(pn)p Fp(\))p 2688 1558 186 4 v 2747 1606 a Fo(p)2781 1589 y Fn(2)2911 1577 y Fs(=)e FB(O)r Fs(\()p FB(p)p Fs(\))k(and)g(this)e(means)289 1688 y(that)f FB(\027)510 1700 y Fo(p)548 1688 y Fs(\()p FB(H)7 b Fs(\()p FB(pn)p Fs(\)\))24 b Fq(\025)e Fs(2.)37 b(W)-7 b(e)28 b(consider)d(then)j(Theorem)2063 1688 y SDict begin H.S end 2063 1688 a 0 TeXcolorgray Fs(3.11)p 0 TeXcolorgray 2212 1634 a SDict begin H.R end 2212 1634 a 2212 1688 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.3.11) cvn H.B /ANN pdfmark end 2212 1688 a 26 w Fs(for)f FB(m)c Fs(=)g(3:)272 1885 y FB(H)7 b Fs(\()p FB(p)422 1851 y Fp(3)459 1885 y FB(n)p Fs(\))24 b(=)662 1829 y FB(H)7 b Fs(\()p FB(n)p Fs(\))p 662 1866 191 4 v 718 1942 a FB(p)760 1918 y Fp(3)881 1885 y Fq(\000)18 b FB(B)1027 1900 y Fo(p)1061 1884 y Fn(2)1093 1900 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(2)1364 1829 y FB(n)1414 1799 y Fp(2)p 1364 1866 88 4 v 1386 1942 a Fs(2)1479 1885 y(+)g FB(O)r Fs(\()p FB(p)p Fs(\))24 b(=)1855 1829 y FB(H)7 b Fs(\()p FB(n)p Fs(\))p 1855 1866 191 4 v 1910 1942 a FB(p)1952 1918 y Fp(3)2073 1885 y Fq(\000)2166 1829 y FB(B)2229 1841 y Fo(p)p Fk(\000)p Fp(3)p 2166 1866 187 4 v 2239 1942 a Fs(3)2363 1885 y FB(n)2413 1851 y Fp(2)2468 1885 y Fs(+)18 b FB(O)r Fs(\()p FB(p)p Fs(\))24 b(=)f Fq(\000)2909 1829 y FB(B)2972 1841 y Fo(p)p Fk(\000)p Fp(3)p 2908 1866 V 2981 1942 a Fs(3)3105 1885 y FB(n)3155 1851 y Fp(2)3210 1885 y Fs(+)18 b FB(O)r Fs(\()p FB(p)p Fs(\))p FB(;)289 2071 y Fs(as)27 b(b)n(y)h(h)n(yp)r(othesis,)e FB(\027)980 2083 y Fo(p)1019 2071 y Fs(\()p FB(H)7 b Fs(\()p FB(n)p Fs(\)\))24 b Fq(\025)g Fs(4.)37 b(The)28 b(RHS)h(has)f(v)-5 b(aluation)25 b(zero)i(if)g FB(p)h Fs(is)f(non-W)-7 b(olstenholme:)34 b(in)28 b(fact,)289 2171 y(using)e(Theorem)857 2171 y SDict begin H.S end 857 2171 a 0 TeXcolorgray Fs(4.1)p 0 TeXcolorgray 964 2117 a SDict begin H.R end 964 2117 a 964 2171 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.4.1) cvn H.B /ANN pdfmark end 964 2171 a 27 w Fs(to)h(justify)g(the)h(Bernoulli)c(n)n(um)n(b)r(er)j (congruence,)571 2355 y Fq(\000)646 2294 y FB(B)709 2309 y Fo(p)743 2292 y Fn(2)775 2309 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(2)p 646 2336 390 4 v 820 2412 a Fs(2)1069 2355 y(=)1207 2294 y FB(B)1270 2309 y Fo(p)1304 2292 y Fn(2)1337 2309 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(2)p 1166 2336 472 4 v 1166 2412 a FB(p)1208 2388 y Fp(2)1245 2412 y Fs(\()p FB(p)19 b Fq(\000)f Fs(1\))g Fq(\000)g Fs(2)1666 2355 y(+)g FB(O)r Fs(\()p FB(p)p Fs(\))24 b(=)2042 2299 y FB(B)2105 2311 y Fo(p)p Fk(\000)p Fp(3)p 2042 2336 187 4 v 2043 2412 a FB(p)18 b Fq(\000)g Fs(3)2256 2355 y(+)g FB(O)r Fs(\()p FB(p)p Fs(\))24 b(=)f Fq(\000)2697 2299 y FB(B)2760 2311 y Fo(p)p Fk(\000)p Fp(3)p 2697 2336 V 2769 2412 a Fs(3)2911 2355 y(+)18 b FB(O)r Fs(\()p FB(p)p Fs(\))p FB(:)0 2475 y SDict begin H.S end 0 2475 a 0 2475 a SDict begin 12 H.A end 0 2475 a 0 2475 a SDict begin [/View [/XYZ H.V]/Dest (Item.8) cvn /DEST pdfmark end 0 2475 a 0 TeXcolorgray 141 2552 a Fs(\(2\))p 0 TeXcolorgray 42 w(By)32 b(h)n(yp)r(othesis,)h (for)f(all)e FB(m)i Fq(2)f Fi(N)t Fs(,)41 b FB(H)7 b Fs(\()p FB(p)1574 2522 y Fo(m)1636 2552 y FB(n)p Fs(\))32 b(=)f FB(O)r Fs(\()p FB(p)p Fs(\),)k(w)n(e)e(then)g(ev)-5 b(aluate)31 b(form)n(ula)f(\()3069 2552 y SDict begin H.S end 3069 2552 a 0 TeXcolorgray Fs(1)p 0 TeXcolorgray 3111 2498 a SDict begin H.R end 3111 2498 a 3111 2552 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.1) cvn H.B /ANN pdfmark end 3111 2552 a Fs(\))j(in)e FB(p)3319 2522 y Fo(j)3354 2552 y FB(n)p Fs(:)47 b Fq(8)p FB(j)36 b Fq(2)c Fi(N)t Fs(,)289 2652 y(where)27 b FB(p)c Fi(-)g FB(n)p Fs(,)361 2795 y FB(H)7 b Fs(\()p FB(p)511 2761 y Fo(m)p Fp(+)p Fo(j)655 2795 y FB(n)p Fs(\))23 b(=)g FB(O)r Fs(\()p FB(p)p Fs(\))84 b(and)361 3015 y FB(H)7 b Fs(\()p FB(p)511 2981 y Fo(m)p Fp(+)p Fo(j)655 3015 y FB(n)p Fs(\))23 b(=)858 2959 y FB(H)7 b Fs(\()p FB(p)1008 2929 y Fo(j)1043 2959 y FB(n)p Fs(\))p 858 2996 267 4 v 939 3072 a FB(p)981 3048 y Fo(m)1153 3015 y Fs(+)1236 2911 y Fo(m)p Fk(\000)p Fp(1)1248 2936 y Fl(X)1247 3115 y Fo(h)p Fp(=2)1435 2911 y Fo(h)1395 2936 y Fl(X)1394 3115 y Fo(k)q Fp(=1)1539 2951 y FB(B)1602 2968 y Fo(p)1636 2951 y Fm(h)1674 2968 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(2)p Fo(k)p 1539 2996 433 4 v 1669 3072 a Fs(2)p FB(k)s(p)1799 3048 y Fo(h)1981 2898 y Fl(\022)2042 2959 y FB(p)2084 2929 y Fo(h)2127 2959 y Fs(\()p FB(p)19 b Fq(\000)f Fs(1\))g Fq(\000)g Fs(1)2166 3072 y(2)p FB(k)j Fq(\000)d Fs(1)2520 2898 y Fl(\023)2581 3015 y FB(p)2623 2981 y Fp(2)p Fo(k)q Fp(\()p Fo(m)p Fk(\000)p Fo(h)p Fp(+)p Fo(j)s Fp(\))2980 3015 y FB(n)3030 2981 y Fp(2)p Fo(k)3122 3015 y Fs(+)g FB(O)r Fs(\()p FB(p)p Fs(\))p FB(:)289 3231 y Fs(W)-7 b(e)29 b(no)n(w)f(aim)e(to)j(pro)n(v)n(e)e (that)i(the)g(term)f(with)g(the)h(lo)n(w)n(est)d FB(p)p Fs(-adic)h(v)-5 b(aluation)26 b(in)i(the)h(sum)f(is)g(obtained)f(for) 289 3331 y FB(h)c Fs(=)f FB(m)d Fq(\000)f Fs(1)p FB(;)c(k)25 b Fs(=)e(1.)36 b(This)27 b(term)f(has)h(v)-5 b(aluation)836 3604 y FB(v)876 3616 y Fp(min)1014 3604 y Fs(=)o FB(\027)1119 3616 y Fo(p)1172 3487 y Fl(\022)1243 3542 y FB(B)1306 3557 y Fo(p)1340 3541 y Fm(m)p Fe(\000)p Fn(1)1468 3557 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(2)p 1243 3585 486 4 v 1370 3661 a Fs(2)p FB(p)1454 3637 y Fo(m)p Fk(\000)p Fp(1)1738 3487 y Fl(\022)1799 3548 y FB(p)1841 3518 y Fo(m)p Fk(\000)p Fp(1)1989 3548 y Fs(\()p FB(p)19 b Fq(\000)f Fs(1\))g Fq(\000)g Fs(1)2070 3661 y(1)2382 3487 y Fl(\023)2443 3604 y FB(p)2485 3570 y Fp(2\()p Fo(j)s Fp(+1\))2689 3604 y FB(n)2739 3570 y Fp(2)2776 3487 y Fl(\023)2860 3604 y Fs(=)1101 3787 y(=)23 b FB(\027)1230 3799 y Fo(p)1282 3719 y Fl(\000)1320 3787 y FB(B)1383 3802 y Fo(p)1417 3786 y Fm(m)p Fe(\000)p Fn(1)1546 3802 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(2)1806 3719 y Fl(\001)1863 3787 y Fs(+)18 b(2)p FB(j)k Fs(+)d(3)e Fq(\000)i FB(m)j Fs(=)1101 3916 y(=)h FB(\027)1230 3928 y Fo(p)1282 3916 y Fs(\()q FB(B)1378 3928 y Fo(p)p Fk(\000)p Fp(3)1501 3916 y Fs(\))c(+)f(2)p FB(j)23 b Fs(+)18 b(3)g Fq(\000)g FB(m)k Fs(=)h(2)p FB(j)g Fs(+)18 b(3)g Fq(\000)g FB(m:)289 4058 y Fs(Once)k(again)f(w)n(e)i(used)g (Theorem)1362 4058 y SDict begin H.S end 1362 4058 a 0 TeXcolorgray Fs(4.1)p 0 TeXcolorgray 1469 4005 a SDict begin H.R end 1469 4005 a 1469 4058 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.4.1) cvn H.B /ANN pdfmark end 1469 4058 a 22 w Fs(and)g(Prop)r(osition)2091 4058 y SDict begin H.S end 2091 4058 a 0 TeXcolorgray Fs(3.2)p 0 TeXcolorgray 2198 4005 a SDict begin H.R end 2198 4005 a 2198 4058 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.3.2) cvn H.B /ANN pdfmark end 2198 4058 a 22 w Fs(to)g(justify)g(the)g (Bernoulli)c(n)n(um)n(b)r(er)j(congruence)468 4244 y Fq(\000)543 4181 y FB(B)606 4197 y Fo(p)640 4180 y Fm(m)p Fe(\000)p Fn(1)768 4197 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(2)p 542 4225 486 4 v 765 4301 a Fs(2)1061 4244 y(=)1207 4181 y FB(B)1270 4197 y Fo(p)1304 4180 y Fm(m)p Fe(\000)p Fn(1)1433 4197 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(2)p 1159 4225 583 4 v 1159 4301 a FB(p)1201 4277 y Fo(m)p Fk(\000)p Fp(1)1349 4301 y Fs(\()p FB(p)c Fq(\000)g Fs(1\))g Fq(\000)g Fs(2)1769 4244 y(+)g FB(O)r Fs(\()p FB(p)p Fs(\))24 b(=)2145 4188 y FB(B)2208 4200 y Fo(p)p Fk(\000)p Fp(3)p 2145 4225 187 4 v 2146 4301 a FB(p)18 b Fq(\000)g Fs(3)2360 4244 y(+)g FB(O)r Fs(\()p FB(p)p Fs(\))24 b(=)e Fq(\000)2800 4188 y FB(B)2863 4200 y Fo(p)p Fk(\000)p Fp(3)p 2800 4225 V 2872 4301 a Fs(3)3015 4244 y(+)c FB(O)r Fs(\()p FB(p)p Fs(\))p FB(;)289 4429 y Fs(from)28 b(whic)n(h)h(follo)n(ws)d (the)k(equiv)-5 b(alence)27 b(of)j(v)-5 b(aluations)27 b(if)i FB(p)g Fs(is)g(non-W)-7 b(olstenholme.)39 b(No)n(w)29 b(w)n(e)g(discuss)f(the)289 4529 y FB(p)p Fs(-adic)g(v)-5 b(aluation)28 b(term)h(b)n(y)h(term)f(in)h(the)h(sum.)44 b(Eac)n(h)29 b(summand)f(ma)n(y)h(b)r(e)h(represen)n(ted)f(as)h(a)g (pair)e(\()p FB(h;)14 b(k)s Fs(\))289 4628 y(as)24 b(co)r(ordinates)f (in)i(the)h(t)n(w)n(o)e(sums)h(form)n(ula)d(with)j(2)d Fq(\024)h FB(h)g Fq(\024)g FB(m)14 b Fq(\000)f Fs(1)25 b(and)g(1)e Fq(\024)f FB(k)k Fq(\024)d FB(h)p Fs(.)36 b(Therefore)24 b(eac)n(h)h(term)289 4728 y(falls)g(in)n(to)i(one)g(of)g (these)h(three)f(groups:)p 0 TeXcolorgray 362 4828 a FB(i)p Fs(\))p 0 TeXcolorgray 41 w(The)h(term)e(corresp)r(onding)f(to)i (\()p FB(h;)14 b(k)s Fs(\))24 b(=)e(\()p FB(m)d Fq(\000)f Fs(1)p FB(;)c Fs(1\),)27 b(discussed)f(ab)r(o)n(v)n(e,)p 0 TeXcolorgray 333 4927 a FB(ii)p Fs(\))p 0 TeXcolorgray 41 w(T)-7 b(erms)26 b(of)i(the)g(forms)e(\()p FB(h;)14 b(k)s Fs(\))27 b(suc)n(h)h(that)g(\()p FB(p)18 b Fq(\000)g Fs(1\))23 b Fq(j)g Fs(2)p FB(k)30 b Fs(\(except)e(the)g(summand)e(in)g (the)i(\()p FB(i)p Fs(\))g(group\),)p 0 TeXcolorgray 304 5027 a FB(iii)p Fs(\))p 0 TeXcolorgray 41 w(The)g(remaining)23 b(terms.)289 5226 y(F)-7 b(or)27 b(the)h(general)d(summand)h(in)g(\() 1374 5226 y SDict begin H.S end 1374 5226 a 0 TeXcolorgray Fs(1)p 0 TeXcolorgray 1416 5173 a SDict begin H.R end 1416 5173 a 1416 5226 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.1) cvn H.B /ANN pdfmark end 1416 5226 a Fs(\))i(w)n(e)f(ha)n(v) n(e:)566 5503 y FB(\027)607 5515 y Fo(p)659 5386 y Fl(\022)730 5439 y FB(B)793 5456 y Fo(p)827 5439 y Fm(h)866 5456 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(2)p Fo(k)p 730 5484 433 4 v 860 5560 a Fs(2)p FB(k)s(p)990 5536 y Fo(h)1173 5386 y Fl(\022)1234 5446 y FB(p)1276 5416 y Fo(h)1319 5446 y Fs(\()p FB(p)18 b Fq(\000)g Fs(1\))g Fq(\000)h Fs(1)1357 5560 y(2)p FB(k)i Fq(\000)d Fs(1)1711 5386 y Fl(\023)1772 5503 y FB(p)1814 5468 y Fp(2)p Fo(k)q Fp(\()p Fo(m)p Fk(\000)p Fo(h)p Fp(+)p Fo(j)s Fp(\))2171 5503 y FB(n)2221 5468 y Fp(2)p Fo(k)2295 5386 y Fl(\023)2379 5503 y Fs(=)668 5770 y(=)23 b FB(\027)797 5782 y Fo(p)849 5703 y Fl(\000)887 5770 y FB(B)950 5787 y Fo(p)984 5770 y Fm(h)1023 5787 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(2)p Fo(k)1320 5703 y Fl(\001)1376 5770 y Fs(+)18 b FB(\027)1500 5782 y Fo(p)1539 5653 y Fl(\022)1600 5714 y FB(p)1642 5684 y Fo(h)1685 5714 y Fs(\()p FB(p)g Fq(\000)g Fs(1\))g Fq(\000)h Fs(1)1723 5827 y(2)p FB(k)i Fq(\000)d Fs(1)2077 5653 y Fl(\023)2157 5770 y Fs(+)g(2)p FB(k)s Fs(\()p FB(m)g Fq(\000)g FB(h)g Fs(+)g FB(j)5 b Fs(\))19 b Fq(\000)f FB(h)g Fq(\000)g FB(\027)3046 5782 y Fo(p)3084 5770 y Fs(\()p FB(k)s Fs(\))p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 8 8 TeXDict begin HPSdict begin 8 7 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -8 -191 a SDict begin H.S end -8 -191 a -8 -191 a SDict begin H.R end -8 -191 a -8 -191 a SDict begin [/View [/XYZ H.V]/Dest (page.8) cvn /DEST pdfmark end -8 -191 a 0 TeXcolorgray 0 -116 a Fp(8)733 b(LEONARDO)23 b(CAR)n(OFIGLIO,)g(LUIGI)f(DE)g(FILPO,)g(ALESSANDR)n(O)g(GAMBINI)p 0 TeXcolorgray 668 83 a Fq(\025)h FB(\027)797 95 y Fo(p)849 16 y Fl(\000)887 83 y FB(B)950 100 y Fo(p)984 83 y Fm(h)1023 100 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(2)p Fo(k)1320 16 y Fl(\001)1376 83 y Fs(+)18 b(2)p FB(k)s Fs(\()p FB(m)g Fq(\000)g FB(h)h Fs(+)f FB(j)5 b Fs(\))18 b Fq(\000)g FB(h)h Fq(\000)f FB(\027)2266 95 y Fo(p)2304 83 y Fs(\()p FB(k)s Fs(\))668 250 y Fq(\025)23 b FB(\027)797 262 y Fo(p)849 182 y Fl(\000)887 250 y FB(B)950 266 y Fo(p)984 250 y Fm(h)1023 266 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(2)p Fo(k)1320 182 y Fl(\001)1376 250 y Fs(+)18 b(2)p FB(k)s Fs(\()p FB(m)g Fq(\000)g FB(h)h Fs(+)f FB(j)5 b Fs(\))18 b Fq(\000)g FB(h)h Fs(+)f(\()p FB(k)j Fq(\000)d FB(\027)2445 262 y Fo(p)2484 250 y Fs(\()p FB(k)s Fs(\))h Fq(\000)f Fs(1\))g Fq(\000)g FB(k)j Fs(+)d(1)668 416 y Fq(\025)23 b FB(\027)797 428 y Fo(p)849 349 y Fl(\000)887 416 y FB(B)950 433 y Fo(p)984 416 y Fm(h)1023 433 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(2)p Fo(k)1320 349 y Fl(\001)1376 416 y Fs(+)18 b(2)p FB(k)s Fs(\()p FB(m)g Fq(\000)g FB(h)h Fs(+)f FB(j)5 b Fs(\))18 b Fq(\000)g FB(h)h Fq(\000)f FB(k)j Fs(+)d(1)668 622 y Fq(\025)23 b FB(\027)797 634 y Fo(p)849 554 y Fl(\000)887 622 y FB(B)950 639 y Fo(p)984 622 y Fm(h)1023 639 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(2)p Fo(k)1320 554 y Fl(\001)1376 622 y Fs(+)18 b(2)p FB(k)1547 530 y Fl(\020)1596 622 y FB(m)h Fq(\000)f FB(h)g Fs(+)g FB(j)23 b Fq(\000)2070 566 y Fs(1)p 2070 603 42 4 v 2070 679 a(2)2122 530 y Fl(\021)2190 622 y Fq(\000)18 b FB(h)g Fs(+)g(1)p FB(;)3655 622 y SDict begin H.S end 3655 622 a 3655 622 a SDict begin 18.45831 H.A end 3655 622 a 3655 622 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.4) cvn /DEST pdfmark end 3655 622 a Fs(\(4\))289 808 y(as)28 b FB(k)f(>)d Fs(1)19 b(+)g FB(\027)737 820 y Fo(p)775 808 y Fs(\()p FB(k)s Fs(\))29 b(for)f FB(p)d Fq(\025)f Fs(5,)k(and)h FB(k)e Fq(\025)d Fs(2.)40 b(No)n(w,)28 b(w)n(e)g(use)h(Corollary)2567 808 y SDict begin H.S end 2567 808 a 0 TeXcolorgray Fs(3.10)p 0 TeXcolorgray 2716 755 a SDict begin H.R end 2716 755 a 2716 808 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.3.10) cvn H.B /ANN pdfmark end 2716 808 a Fs(:)37 b FB(\027)2817 820 y Fo(p)2870 741 y Fl(\000)2908 808 y FB(B)2971 825 y Fo(p)3005 809 y Fm(h)3043 825 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(2)p Fo(k)3340 741 y Fl(\001)3403 808 y Fq(\025)24 b(\000)p Fs(1)k(and)289 919 y FB(\027)330 931 y Fo(p)382 851 y Fl(\000)420 919 y FB(B)483 936 y Fo(p)517 919 y Fm(h)556 936 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(2)p Fo(k)853 851 y Fl(\001)914 919 y Fs(=)22 b Fq(\000)p Fs(1)27 b(if)g(and)h(only)e(if)h(\()p FB(p)18 b Fq(\000)g Fs(1\))23 b Fq(j)g Fs(2)p FB(k)s Fs(.)289 1021 y(If)28 b(the)g(terms)e(b)r(elong)g(to)i(group)e(\()p FB(ii)p Fs(\),)i(\()p FB(p)18 b Fq(\000)g Fs(1\))23 b Fq(j)g Fs(2)p FB(k)s Fs(,)k(then)h(\()2198 1021 y SDict begin H.S end 2198 1021 a 0 TeXcolorgray Fs(4)p 0 TeXcolorgray 2240 967 a SDict begin H.R end 2240 967 a 2240 1021 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.4) cvn H.B /ANN pdfmark end 2240 1021 a Fs(\))g(is)f(greater)e(than)1280 1210 y(2)p FB(k)1368 1118 y Fl(\020)1417 1210 y FB(m)19 b Fq(\000)f FB(h)g Fs(+)g FB(j)23 b Fq(\000)1891 1154 y Fs(1)p 1891 1191 42 4 v 1891 1267 a(2)1943 1118 y Fl(\021)2011 1210 y Fq(\000)18 b FB(h)1216 1410 y Fq(\025)o Fs(\()p FB(p)h Fq(\000)f Fs(1\)\()p FB(m)g Fq(\000)g FB(h)h Fs(+)f FB(j)23 b Fq(\000)2036 1354 y Fs(1)p 2036 1391 V 2036 1467 a(2)2088 1410 y(\))18 b Fq(\000)g FB(m)h Fs(+)f(1)1216 1610 y Fq(\025)o Fs(\()p FB(p)h Fq(\000)f Fs(1\)\(1)g(+)g FB(j)23 b Fq(\000)1855 1554 y Fs(1)p 1855 1591 V 1855 1667 a(2)1907 1610 y(\))c Fq(\000)f FB(m)g Fs(+)g(1)23 b FB(>)f(v)2407 1622 y Fp(min)2522 1610 y FB(:)289 1788 y Fs(Otherwise,)i(if)i(the)g(terms)f(b)r(elong)f(to)i(group)f(\()p FB(iii)p Fs(\),)g(the)i(inequalit)n(y)22 b FB(h)h(<)g(m)15 b Fq(\000)f Fs(1)26 b(is)f(strict,)g(b)r(earing)f(in)h(mind)289 1888 y(that)j(for)f(group)f(\()p FB(iii)p Fs(\))h(w)n(e)g(ha)n(v)n(e)g (\()p FB(h;)14 b(k)s Fs(\))23 b Fq(6)p Fs(=)g(\()p FB(m)18 b Fq(\000)g Fs(1)p FB(;)c Fs(1\))27 b(and)h FB(k)e Fq(\025)c Fs(2.)37 b(Then)27 b(\()2683 1888 y SDict begin H.S end 2683 1888 a 0 TeXcolorgray Fs(4)p 0 TeXcolorgray 2725 1835 a SDict begin H.R end 2725 1835 a 2725 1888 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.4) cvn H.B /ANN pdfmark end 2725 1888 a Fs(\))h(is)f(greater)e(than)1410 2078 y(2)p FB(k)1498 1986 y Fl(\020)1547 2078 y FB(m)19 b Fq(\000)f FB(h)g Fs(+)g FB(j)23 b Fq(\000)2021 2022 y Fs(1)p 2021 2059 42 4 v 2021 2135 a(2)2073 1986 y Fl(\021)2141 2078 y Fq(\000)18 b FB(h)g Fs(+)g(1)1346 2278 y FB(>)o Fs(2)p FB(k)1498 2185 y Fl(\020)1557 2221 y Fs(1)p 1557 2259 V 1557 2335 a(2)1627 2278 y(+)g FB(j)1749 2185 y Fl(\021)1817 2278 y Fq(\000)g FB(m)h Fs(+)f(2)1346 2430 y FB(>)o Fs(2\(1)g(+)g FB(j)5 b Fs(\))19 b Fq(\000)f FB(m)g Fs(+)g(1)1346 2555 y(=)o(2)p FB(j)23 b Fs(+)18 b(3)g Fq(\000)g FB(m)23 b Fs(=)g FB(v)1959 2567 y Fp(min)2073 2555 y FB(:)289 2706 y Fs(Therefore)33 b(w)n(e)h(\014nd)h(that)f(the)h(sum)f(has)f(v)-5 b(aluation)32 b FB(v)2056 2718 y Fp(min)2171 2706 y Fs(,)k(whic)n(h)d (is)g(negativ)n(e)g(for)g(su\016cien)n(tly)g(large)f FB(m)p Fs(,)289 2805 y(th)n(us)870 2954 y FB(\027)911 2966 y Fo(p)963 2887 y Fl(\000)1001 2954 y FB(H)7 b Fs(\()p FB(p)1151 2920 y Fo(j)1186 2954 y FB(n)p Fs(\))1268 2887 y Fl(\001)1325 2954 y Fq(\000)18 b FB(m)23 b Fs(=)f FB(\027)1632 2966 y Fo(p)1685 2837 y Fl(\022)1756 2898 y FB(H)7 b Fs(\()p FB(p)1906 2868 y Fo(j)1941 2898 y FB(n)p Fs(\))p 1756 2935 267 4 v 1837 3011 a FB(p)1879 2987 y Fo(m)2033 2837 y Fl(\023)2117 2954 y Fs(=)22 b FB(v)2244 2966 y Fp(min)2382 2954 y Fs(=)h(2)p FB(j)f Fs(+)c(3)g Fq(\000)g FB(m;)289 3133 y Fs(that)28 b(means)1484 3248 y FB(\027)1525 3260 y Fo(p)1577 3181 y Fl(\000)1615 3248 y FB(H)7 b Fs(\()p FB(p)1765 3214 y Fo(j)1800 3248 y FB(n)p Fs(\))1882 3181 y Fl(\001)1943 3248 y Fs(=)23 b(2)p FB(j)f Fs(+)c(3)p FB(:)0 3311 y SDict begin H.S end 0 3311 a 0 3311 a SDict begin 12 H.A end 0 3311 a 0 3311 a SDict begin [/View [/XYZ H.V]/Dest (Item.9) cvn /DEST pdfmark end 0 3311 a 0 TeXcolorgray 141 3382 a Fs(\(3\))p 0 TeXcolorgray 42 w(If)26 b(instead)f(there)h(exists)f FB(m)e Fq(2)g Fi(N)36 b Fs(suc)n(h)26 b(that)g FB(\027)1759 3394 y Fo(p)1797 3382 y Fs(\()p FB(H)7 b Fs(\()p FB(p)1979 3351 y Fo(m)2043 3382 y FB(n)p Fs(\)\))23 b Fq(\024)g Fs(2,)j(then)g(b)n(y)g (the)g(descen)n(t)g(phenomenon)f(there)289 3483 y(exists)g FB(M)18 b Fq(2)23 b Fi(N)37 b Fs(suc)n(h)27 b(that)g FB(\027)1188 3495 y Fo(p)1226 3483 y Fs(\()p FB(H)7 b Fs(\()p FB(p)1408 3453 y Fo(M)1482 3483 y FB(n)p Fs(\)\))24 b(=)e(0.)36 b(Then)27 b(again)e(ev)-5 b(aluating)24 b(form)n(ula)g(\() 2979 3483 y SDict begin H.S end 2979 3483 a 0 TeXcolorgray Fs(1)p 0 TeXcolorgray 3021 3430 a SDict begin H.R end 3021 3430 a 3021 3483 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.1) cvn H.B /ANN pdfmark end 3021 3483 a Fs(\))j(in)f FB(H)7 b Fs(\()p FB(p)3326 3453 y Fo(M)3413 3483 y FB(p)3455 3453 y Fo(j)3490 3483 y FB(n)p Fs(\))27 b(with)289 3584 y FB(m)c Fs(=)f FB(M)37 b Fs(in)26 b FB(p)728 3554 y Fo(j)763 3584 y FB(n)p Fs(,)i(where)f FB(p)c Fi(-)g FB(n)p Fs(,)k(w)n(e)h(ha)n(v)n(e)340 3817 y FB(H)7 b Fs(\()p FB(p)490 3782 y Fo(M)578 3817 y FB(p)620 3782 y Fo(j)654 3817 y FB(n)p Fs(\))24 b(=)857 3760 y FB(H)7 b Fs(\()p FB(p)1007 3730 y Fo(j)1042 3760 y FB(n)p Fs(\))p 857 3798 267 4 v 933 3874 a FB(p)975 3850 y Fo(M)1152 3817 y Fs(+)1236 3713 y Fo(M)f Fk(\000)p Fp(1)1253 3738 y Fl(X)1251 3917 y Fo(h)p Fp(=2)1445 3713 y Fo(h)1404 3738 y Fl(X)1404 3917 y Fo(k)q Fp(=1)1548 3753 y FB(B)1611 3770 y Fo(p)1645 3753 y Fm(h)1684 3770 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(2)p Fo(k)p 1548 3798 433 4 v 1679 3874 a Fs(2)p FB(k)s(p)1809 3850 y Fo(h)1991 3700 y Fl(\022)2052 3760 y FB(p)2094 3730 y Fo(h)2137 3760 y Fs(\()p FB(p)18 b Fq(\000)g Fs(1\))h Fq(\000)f Fs(1)2176 3874 y(2)p FB(k)i Fq(\000)e Fs(1)2529 3700 y Fl(\023)2590 3817 y FB(p)2632 3782 y Fp(2)p Fo(k)q Fp(\()p Fo(M)6 b Fk(\000)p Fo(h)p Fp(+)p Fo(j)s Fp(\))3000 3817 y FB(n)3050 3782 y Fp(2)p Fo(k)3142 3817 y Fs(+)18 b FB(O)r Fs(\()p FB(p)p Fs(\))p FB(;)289 4046 y Fs(where)27 b(the)g(LHS)h(has)f(v)-5 b(aluation)25 b Fq(\000)p FB(j)5 b Fs(,)27 b(and)g(b)n(y)g(the)h(same)e(reasoning)e(as)j(in)f(case)h (\(2\))g(the)h(RHS)g(is)e(a)h(sum)g(of)289 4146 y(t)n(w)n(o)d(terms)g (of)h(v)-5 b(aluation)23 b(resp)r(ectiv)n(ely)f FB(\027)1615 4158 y Fo(p)1654 4146 y Fs(\()p FB(H)7 b Fs(\()p FB(p)1836 4116 y Fo(j)1871 4146 y FB(n)p Fs(\)\))14 b Fq(\000)g FB(M)33 b Fs(and)25 b(2)p FB(j)18 b Fs(+)c(3)g Fq(\000)g FB(M)9 b Fs(.)33 b(No)n(w)25 b(surely)f(b)r(oth)h(v)-5 b(aluation)289 4246 y(cannot)26 b(exceed)h Fq(\000)p FB(j)5 b Fs(,)27 b(since)f(then)h(the)h(LHS)f(and)g(RHS)h(w)n(ould)d (then)j(ha)n(v)n(e)e(di\013eren)n(t)g(v)-5 b(aluations.)34 b(Th)n(us,)27 b(w)n(e)289 4345 y(ha)n(v)n(e)f(the)i(follo)n(wing)c (cases:)p 0 TeXcolorgray 381 4445 a Fj(\210)p 0 TeXcolorgray 42 w Fs(If)k(3)p FB(j)23 b Fs(+)18 b(3)23 b FB(>)f(M)9 b Fs(,)27 b(then)i FB(\027)1252 4457 y Fo(p)1290 4445 y Fs(\()p FB(H)7 b Fs(\()p FB(p)1472 4415 y Fo(j)1507 4445 y FB(n)p Fs(\)\))24 b(=)e FB(M)27 b Fq(\000)18 b FB(j)5 b Fs(.)p 0 TeXcolorgray 381 4545 a Fj(\210)p 0 TeXcolorgray 42 w Fs(If)28 b FB(\027)588 4557 y Fo(p)626 4545 y Fs(\()p FB(H)7 b Fs(\()p FB(p)808 4515 y Fo(j)843 4545 y FB(n)p Fs(\)\))24 b FB(>)f(M)k Fq(\000)18 b FB(j)5 b Fs(,)27 b(then)h(3)p FB(j)23 b Fs(+)18 b(3)23 b(=)g FB(M)9 b Fs(.)p 0 TeXcolorgray 381 4646 a Fj(\210)p 0 TeXcolorgray 42 w Fs(If)28 b(3)p FB(j)23 b Fs(+)18 b(3)23 b Fq(\024)f FB(M)36 b Fs(and)28 b FB(\027)1201 4658 y Fo(p)1239 4646 y Fs(\()p FB(H)7 b Fs(\()p FB(p)1421 4616 y Fo(j)1456 4646 y FB(n)p Fs(\)\))24 b Fq(\024)e FB(M)27 b Fq(\000)18 b FB(j)5 b Fs(,)28 b(then)g FB(\027)2192 4658 y Fo(p)2231 4646 y Fs(\()p FB(H)7 b Fs(\()p FB(p)2413 4616 y Fo(j)2448 4646 y FB(n)p Fs(\)\))23 b(=)g(2)p FB(j)g Fs(+)18 b(3.)289 4746 y(Therefore,)26 b(b)n(y)h(rearranging)d(the)k (results,)e(w)n(e)h(conclude)g(the)h(pro)r(of.)3696 4753 y Ff(\003)0 4826 y SDict begin H.S end 0 4826 a 0 4826 a SDict begin 12 H.A end 0 4826 a 0 4826 a SDict begin [/View [/XYZ H.V]/Dest (subsection.4.2) cvn /DEST pdfmark end 0 4826 a 99 x Fs(4.2.)41 b FA(Pro)s(of)47 b(of)h(Theorem)996 4925 y SDict begin H.S end 996 4925 a 0 TeXcolorgray FA(2.6)p 0 TeXcolorgray 1119 4872 a SDict begin H.R end 1119 4872 a 1119 4925 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.2.6) cvn H.B /ANN pdfmark end 1119 4925 a FA(.)40 b Fs(Using)h(the)g(same)f(argumen)n(ts)f(w)n(e)i(can)h(pro)n(v)n(e)d (the)j(analogous)c(theorem)i(for)0 5025 y(W)-7 b(olstenholme)24 b(primes.)p 0 TeXcolorgray 0 5197 a Fg(Pr)l(o)l(of)31 b(of)f(The)l(or)l(em)665 5197 y SDict begin H.S end 665 5197 a 0 TeXcolorgray Fg(2.6)p 0 TeXcolorgray 774 5144 a SDict begin H.R end 774 5144 a 774 5197 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.2.6) cvn H.B /ANN pdfmark end 774 5197 a Fg(.)p 0 TeXcolorgray 0 5239 a SDict begin H.S end 0 5239 a 0 5239 a SDict begin 12 H.A end 0 5239 a 0 5239 a SDict begin [/View [/XYZ H.V]/Dest (Item.10) cvn /DEST pdfmark end 0 5239 a 0 TeXcolorgray 141 5322 a Fs(\(1\))p 0 TeXcolorgray 42 w(Again,)25 b(using)h(the)i(same)d (reasoning)f(as)j(in)f(the)h(previous)e(Theorem,)h(w)n(e)g(ev)-5 b(aluate)26 b(form)n(ula)e(\()3315 5322 y SDict begin H.S end 3315 5322 a 0 TeXcolorgray Fs(1)p 0 TeXcolorgray 3357 5269 a SDict begin H.R end 3357 5269 a 3357 5322 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.1) cvn H.B /ANN pdfmark end 3357 5322 a Fs(\))j(in)f FB(m)d Fs(=)g(3:)346 5526 y FB(H)7 b Fs(\()p FB(p)496 5492 y Fp(3)533 5526 y FB(n)p Fs(\))24 b(=)736 5470 y FB(H)7 b Fs(\()p FB(n)p Fs(\))p 736 5507 191 4 v 792 5583 a FB(p)834 5559 y Fp(3)955 5526 y Fq(\000)18 b FB(B)1101 5541 y Fo(p)1135 5524 y Fn(2)1167 5541 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(2)1438 5470 y FB(n)1488 5440 y Fp(2)p 1438 5507 88 4 v 1460 5583 a Fs(2)1553 5526 y(+)g FB(O)r Fs(\()p FB(p)p Fs(\))24 b(=)1929 5470 y FB(H)7 b Fs(\()p FB(n)p Fs(\))p 1929 5507 191 4 v 1984 5583 a FB(p)2026 5559 y Fp(3)2147 5526 y Fs(+)2240 5470 y FB(B)2303 5482 y Fo(p)p Fk(\000)p Fp(3)p 2240 5507 187 4 v 2313 5583 a Fs(3)2437 5526 y FB(n)2487 5492 y Fp(2)2542 5526 y Fs(+)18 b FB(O)r Fs(\()p FB(p)p Fs(\))24 b(=)2918 5470 y FB(H)7 b Fs(\()p FB(n)p Fs(\))p 2918 5507 191 4 v 2973 5583 a FB(p)3015 5559 y Fp(3)3136 5526 y Fs(+)18 b FB(O)r Fs(\()p FB(p)p Fs(\))p FB(;)289 5716 y Fs(where)24 b(the)g(RHS)h(has)f(v)-5 b(aluation)22 b(zero.)35 b(W)-7 b(e)25 b(note)f(that)h(this)e(fact)i (pro)n(v)n(es)e(the)h(descen)n(t)h(phenomenon)e(in)g(the)289 5816 y(case)j FB(m)d Fs(=)g(3)k(for)g(W)-7 b(olstenholme)25 b(primes.)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 9 9 TeXDict begin HPSdict begin 9 8 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -8 -191 a SDict begin H.S end -8 -191 a -8 -191 a SDict begin H.R end -8 -191 a -8 -191 a SDict begin [/View [/XYZ H.V]/Dest (page.9) cvn /DEST pdfmark end -8 -191 a 0 TeXcolorgray 155 -116 a Fo(p)p Fp(-ADIC)30 b(V)-7 b(ALUA)i(TION)29 b(OF)g(HARMONIC)h(SUMS)e(AND)h(THEIR)g (CONNECTIONS)h(WITH)f(W)n(OLSTENHOLME)f(PRIMES)122 b(9)p 0 TeXcolorgray 0 0 a SDict begin H.S end 0 0 a 0 0 a SDict begin 12 H.A end 0 0 a 0 0 a SDict begin [/View [/XYZ H.V]/Dest (Item.11) cvn /DEST pdfmark end 0 0 a 0 TeXcolorgray 141 83 a Fs(\(2\))p 0 TeXcolorgray 42 w(As)27 b(b)r(efore,)h(w)n(e)f(ev)-5 b(aluate)26 b(form)n(ula)f(\()1468 83 y SDict begin H.S end 1468 83 a 0 TeXcolorgray Fs(1)p 0 TeXcolorgray 1510 30 a SDict begin H.R end 1510 30 a 1510 83 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.1) cvn H.B /ANN pdfmark end 1510 83 a Fs(\))j(in)e FB(p)1708 53 y Fo(j)1743 83 y FB(n)p Fs(,)i(where)f FB(p)c Fi(-)g FB(n)p Fs(:)361 227 y FB(H)7 b Fs(\()p FB(p)511 193 y Fo(m)p Fp(+)p Fo(j)655 227 y FB(n)p Fs(\))23 b(=)g FB(O)r Fs(\()p FB(p)p Fs(\))84 b(and)361 447 y FB(H)7 b Fs(\()p FB(p)511 413 y Fo(m)p Fp(+)p Fo(j)655 447 y FB(n)p Fs(\))23 b(=)858 391 y FB(H)7 b Fs(\()p FB(p)1008 361 y Fo(j)1043 391 y FB(n)p Fs(\))p 858 428 267 4 v 939 504 a FB(p)981 480 y Fo(m)1153 447 y Fs(+)1236 343 y Fo(m)p Fk(\000)p Fp(1)1248 368 y Fl(X)1247 547 y Fo(h)p Fp(=2)1435 343 y Fo(h)1395 368 y Fl(X)1394 547 y Fo(k)q Fp(=1)1539 383 y FB(B)1602 400 y Fo(p)1636 383 y Fm(h)1674 400 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(2)p Fo(k)p 1539 428 433 4 v 1669 504 a Fs(2)p FB(k)s(p)1799 480 y Fo(h)1981 330 y Fl(\022)2042 391 y FB(p)2084 361 y Fo(h)2127 391 y Fs(\()p FB(p)19 b Fq(\000)f Fs(1\))g Fq(\000)g Fs(1)2166 504 y(2)p FB(k)j Fq(\000)d Fs(1)2520 330 y Fl(\023)2581 447 y FB(p)2623 413 y Fp(2)p Fo(k)q Fp(\()p Fo(m)p Fk(\000)p Fo(h)p Fp(+)p Fo(j)s Fp(\))2980 447 y FB(n)3030 413 y Fp(2)p Fo(k)3122 447 y Fs(+)g FB(O)r Fs(\()p FB(p)p Fs(\))p FB(:)289 663 y Fs(Di\013eren)n(tly)k(from)h(the)i(pro)r(of)e(of)h (Theorem)1681 663 y SDict begin H.S end 1681 663 a 0 TeXcolorgray Fs(2.5)p 0 TeXcolorgray 1788 610 a SDict begin H.R end 1788 610 a 1788 663 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.2.5) cvn H.B /ANN pdfmark end 1788 663 a 24 w Fs(the)g(minim)n(um)c(v)-5 b(aluation)22 b(term)h(in)g(the)i (sum)e(is)g(not)h(kno)n(wn,)289 763 y(but)i(the)g(lo)n(w)n(est)e(v)-5 b(aluation)23 b(of)j(eac)n(h)f(summand)f(is)g FB(v)1964 775 y Fp(min)2102 763 y Fq(\025)e Fs(2)p FB(j)d Fs(+)c(4)f Fq(\000)g FB(m)26 b Fs(according)d(to)j(the)g(same)e(argumen)n(t)289 862 y(used)j(in)g(the)h(previous)e(pro)r(of.)36 b(Th)n(us,)27 b(since)g(the)h(LHS)g(has)f(p)r(ositiv)n(e)e(v)-5 b(aluation,)25 b(w)n(e)i(ha)n(v)n(e)1363 1054 y FB(\027)1404 1066 y Fo(p)1457 937 y Fl(\022)1528 998 y FB(H)7 b Fs(\()p FB(p)1678 968 y Fo(j)1712 998 y FB(n)p Fs(\))p 1528 1035 267 4 v 1609 1111 a FB(p)1651 1087 y Fo(m)1805 937 y Fl(\023)1889 1054 y Fq(\025)22 b Fs(2)p FB(j)h Fs(+)18 b(4)g Fq(\000)g FB(m;)289 1241 y Fs(otherwise,)25 b(thanks)j(to)f(ultrametric)d (inequalit)n(y)-7 b(,)25 b(w)n(e)i(w)n(ould)f(ha)n(v)n(e)1349 1383 y FB(\027)1390 1395 y Fo(p)1442 1316 y Fl(\000)1480 1383 y FB(H)7 b Fs(\()p FB(p)1630 1349 y Fo(m)1707 1383 y FB(p)1749 1349 y Fo(j)1783 1383 y FB(n)p Fs(\))1865 1316 y Fl(\001)1927 1383 y Fs(=)22 b(2)p FB(j)h Fs(+)18 b(4)g Fq(\000)g FB(m)289 1525 y Fs(that)28 b(con)n(tradicts)d(the)j(h)n (yp)r(othesis)e(for)h(su\016cien)n(tly)f(large)f FB(m)p Fs(.)289 1624 y(If,)h(instead,)e(there)h(exists)f FB(m)f Fq(2)g Fi(N)35 b Fs(suc)n(h)25 b(that)g FB(\027)1799 1636 y Fo(p)1838 1624 y Fs(\()p FB(H)7 b Fs(\()p FB(p)2020 1594 y Fo(m)2083 1624 y FB(n)p Fs(\)\))23 b Fq(\024)g Fs(3)i(then)g(b)n(y)g(descenden)n(t)g(phenomenon)f(there)289 1726 y(exists)i FB(M)32 b Fq(2)23 b Fi(N)38 b Fs(suc)n(h)27 b(that)h FB(\027)1205 1738 y Fo(p)1243 1726 y Fs(\()p FB(H)7 b Fs(\()p FB(p)1425 1696 y Fo(M)1499 1726 y FB(n)p Fs(\)\))23 b(=)g(0.)37 b(Therefore,)26 b(w)n(e)h(ha)n(v)n(e:)1498 1874 y FB(\027)1539 1886 y Fo(p)1578 1874 y Fs(\()p FB(H)7 b Fs(\()p FB(p)1760 1840 y Fo(M)f Fp(+)p Fo(j)1915 1874 y FB(n)p Fs(\)\))24 b(=)e Fq(\000)p FB(j:)289 2017 y Fs(Then)39 b(again,)g(ev)-5 b(aluating)36 b(form)n(ula)g(\()1533 2017 y SDict begin H.S end 1533 2017 a 0 TeXcolorgray Fs(1)p 0 TeXcolorgray 1575 1964 a SDict begin H.R end 1575 1964 a 1575 2017 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.1) cvn H.B /ANN pdfmark end 1575 2017 a Fs(\))j(for)f FB(m)k Fs(=)f FB(M)48 b Fs(in)38 b FB(p)2284 1987 y Fo(j)2319 2017 y FB(n)p Fs(,)j(where)e FB(p)i Fi(-)h FB(n)p Fs(,)g(w)n(e)c(ha)n (v)n(e)g(the)h(follo)n(wing)289 2117 y(congruence:)363 2420 y FB(H)7 b Fs(\()p FB(p)513 2385 y Fo(M)601 2420 y FB(p)643 2385 y Fo(j)677 2420 y FB(n)p Fs(\))24 b(=)857 2363 y FB(H)7 b Fs(\()p FB(p)1007 2333 y Fo(j)1042 2363 y FB(n)p Fs(\))p 857 2400 267 4 v 933 2477 a FB(p)975 2453 y Fo(M)1152 2420 y Fs(+)1236 2316 y Fo(M)f Fk(\000)p Fp(1)1253 2341 y Fl(X)1251 2519 y Fo(h)p Fp(=2)1445 2316 y Fo(h)1404 2341 y Fl(X)1404 2519 y Fo(k)q Fp(=1)1548 2356 y FB(B)1611 2373 y Fo(p)1645 2356 y Fm(h)1684 2373 y Fp(\()p Fo(p)p Fk(\000)p Fp(1\))p Fk(\000)p Fp(2)p Fo(k)p 1548 2400 433 4 v 1679 2477 a Fs(2)p FB(k)s(p)1809 2453 y Fo(h)1991 2303 y Fl(\022)2052 2363 y FB(p)2094 2333 y Fo(h)2137 2363 y Fs(\()p FB(p)18 b Fq(\000)g Fs(1\))h Fq(\000)f Fs(1)2176 2477 y(2)p FB(k)i Fq(\000)e Fs(1)2529 2303 y Fl(\023)2590 2420 y FB(p)2632 2385 y Fp(2)p Fo(k)q Fp(\()p Fo(M)6 b Fk(\000)p Fo(h)p Fp(+)p Fo(j)s Fp(\))3000 2420 y FB(n)3050 2385 y Fp(2)p Fo(k)3142 2420 y Fs(+)18 b FB(O)r Fs(\()p FB(p)p Fs(\))783 2674 y(=)880 2618 y(\001)p 857 2655 116 4 v 857 2731 a FB(p)899 2707 y Fo(M)1001 2674 y Fs(+)g(\006)p FB(;)289 2855 y Fs(sa)n(y)-7 b(.)36 b(In)27 b(this)g(form)n(ula)e(the)j FB(p)p Fs(-adic)e(v)-5 b(aluation)25 b(of)i(the)h(three)g(terms)e(are:)p 0 TeXcolorgray 381 2955 a Fq(\017)p 0 TeXcolorgray 41 w FB(\027)505 2967 y Fo(p)543 2955 y Fs(\()p FB(H)7 b Fs(\()p FB(p)725 2925 y Fo(M)813 2955 y FB(p)855 2925 y Fo(j)890 2955 y FB(n)p Fs(\)\))23 b(=)g Fq(\000)p FB(j)5 b Fs(,)p 0 TeXcolorgray 381 3107 a Fq(\017)p 0 TeXcolorgray 41 w FB(\027)505 3119 y Fo(p)557 2990 y Fl(\022)651 3051 y Fs(\001)p 628 3088 V 628 3164 a FB(p)670 3140 y Fo(M)754 2990 y Fl(\023)838 3107 y Fs(=)23 b FB(\027)967 3119 y Fo(p)1005 3107 y Fs(\(\001\))c Fq(\000)f FB(M)9 b Fs(,)p 0 TeXcolorgray 381 3257 a Fq(\017)p 0 TeXcolorgray 41 w FB(\027)505 3269 y Fo(p)543 3257 y Fs(\(\006\))24 b Fq(\025)f Fs(2)p FB(j)g Fs(+)18 b(4)g Fq(\000)g FB(M)9 b Fs(.)289 3356 y(In)31 b(order)f(to)i(complete)d(the)j(pro)r(of,)g(w)n (e)f(will)e(use)i(the)h FB(p)p Fs(-adic)d(v)-5 b(aluation)29 b(inequalit)n(y)f(to)j(\014nd)h(a)f(b)r(ound)h(for)289 3456 y FB(\027)330 3468 y Fo(p)368 3456 y Fs(\(\001\).)61 b(Ho)n(w)n(ev)n(er,)35 b(in)f(con)n(trast)g(to)h(the)h(situation)d(in)h (Theorem)2461 3456 y SDict begin H.S end 2461 3456 a 0 TeXcolorgray Fs(2.5)p 0 TeXcolorgray 2568 3402 a SDict begin H.R end 2568 3402 a 2568 3456 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.2.5) cvn H.B /ANN pdfmark end 2568 3456 a 35 w Fs(case)g(\(3\),)j(there)e(is)f(not)i(the)f(exact)289 3556 y(v)-5 b(aluation)25 b(of)i FB(\027)786 3568 y Fo(p)825 3556 y Fs(\(\006\).)37 b(Hence,)28 b(the)g(p)r(ossibilities)23 b(are:)793 3697 y FB(i)p Fs(\))83 b FB(\027)978 3709 y Fo(p)1017 3697 y Fs(\(\001\))19 b Fq(\000)f FB(M)32 b Fs(=)22 b Fq(\000)p FB(j)1055 b(\027)2647 3709 y Fo(p)2686 3697 y Fs(\(\006\))23 b FB(>)g Fq(\000)p FB(j)765 3827 y(ii)p Fs(\))82 b FB(\027)978 3839 y Fo(p)1017 3827 y Fs(\(\001\))19 b Fq(\000)f FB(M)32 b(>)22 b Fq(\000)p FB(j)1055 b(\027)2647 3839 y Fo(p)2686 3827 y Fs(\(\006\))23 b(=)g Fq(\000)p FB(j)736 3956 y(iii)p Fs(\))82 b FB(\027)978 3968 y Fo(p)1017 3956 y Fs(\(\001\))19 b Fq(\000)f FB(M)32 b Fs(=)22 b FB(\027)1493 3968 y Fo(p)1532 3956 y Fs(\(\006\))h Fq(\024)g(\000)p FB(j)289 4103 y Fs(W)-7 b(e)21 b(com)n(bine)d(these)i (results)f(with)h(the)h(inequalit)n(y)c(for)j FB(\027)2062 4115 y Fo(p)2101 4103 y Fs(\(\006\))h(in)f(order)f(to)h(estimate)f FB(\027)3007 4115 y Fo(p)3045 4103 y Fs(\(\001\).)36 b(It)21 b(is)e(imp)r(ortan)n(t)289 4203 y(to)28 b(notice)g(that)h(the)g (condition)e FB(\027)1369 4215 y Fo(p)1407 4203 y Fs(\(\006\))f FB(>)f Fq(\000)p FB(j)33 b Fs(do)r(es)c(not)f(pro)n(vide)f(information) e(com)n(bined)i(with)h(the)h(other)289 4302 y(inequalit)n(y)24 b(concerning)h FB(\027)1130 4314 y Fo(p)1169 4302 y Fs(\(\006\).)472 4444 y FB(i:a)p Fs(\))83 b FB(\027)724 4456 y Fo(p)763 4444 y Fs(\(\001\))23 b(=)g FB(M)k Fq(\000)18 b FB(j)28 b(<)23 b Fs(2)p FB(j)g Fs(+)18 b(4)637 b(if)27 b FB(M)k(<)23 b Fs(3)p FB(j)g Fs(+)18 b(4)27 b(and)h(\()p FB(i)p Fs(\))f(holds,)481 4574 y FB(i:b)p Fs(\))82 b FB(\027)724 4586 y Fo(p)763 4574 y Fs(\(\001\))23 b(=)g FB(M)k Fq(\000)18 b FB(j)28 b Fq(\025)23 b Fs(2)p FB(j)g Fs(+)18 b(4)637 b(if)27 b FB(M)k Fq(\025)23 b Fs(3)p FB(j)g Fs(+)18 b(4)27 b(and)h(\()p FB(i)p Fs(\))f(holds,)511 4703 y FB(ii)p Fs(\))82 b FB(\027)724 4715 y Fo(p)763 4703 y Fs(\(\001\))23 b FB(>)g(M)k Fq(\000)18 b FB(j)28 b(>)23 b Fs(2)p FB(j)g Fs(+)18 b(4)637 b(if)27 b FB(M)k Fq(\025)23 b Fs(3)p FB(j)g Fs(+)18 b(4)27 b(and)h(\()p FB(ii)p Fs(\))f(holds,)482 4833 y FB(iii)p Fs(\))82 b FB(\027)724 4845 y Fo(p)763 4833 y Fs(\(\001\))23 b(=)g FB(\027)1048 4845 y Fo(p)1086 4833 y Fs(\(\006\))c(+)f FB(M)32 b Fq(\025)23 b Fs(2)p FB(j)g Fs(+)18 b(4)472 b(if)27 b FB(M)k Fq(\025)23 b Fs(3)p FB(j)g Fs(+)18 b(4)27 b(and)h(\()p FB(iii)p Fs(\))f(holds.)289 4975 y(This)f(yields)g(the)i (desired)e(result.)3696 4978 y Ff(\003)0 5069 y SDict begin H.S end 0 5069 a 0 5069 a SDict begin 12 H.A end 0 5069 a 0 5069 a SDict begin [/View [/XYZ H.V]/Dest (section.5) cvn /DEST pdfmark end 0 5069 a 985 5168 a Fs(5.)42 b Fr(Conclusions)30 b(and)h(fur)-6 b(ther)33 b(developments)100 5318 y Fs(W)-7 b(e)22 b(highligh)n(t)c(the)k(fact)g(that)f(the)h (existences)e(of)i(pairs)e(\()p FB(n;)14 b(p)p Fs(\))21 b(satisfying)e(case)i(\(2\))h(in)e(Theorems)3195 5318 y SDict begin H.S end 3195 5318 a 0 TeXcolorgray Fs(2.5)p 0 TeXcolorgray 3302 5264 a SDict begin H.R end 3302 5264 a 3302 5318 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.2.5) cvn H.B /ANN pdfmark end 3302 5318 a 21 w Fs(or)3418 5318 y SDict begin H.S end 3418 5318 a 0 TeXcolorgray Fs(2.6)p 0 TeXcolorgray 3525 5264 a SDict begin H.R end 3525 5264 a 3525 5318 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.2.6) cvn H.B /ANN pdfmark end 3525 5318 a 21 w Fs(w)n(ould)0 5417 y(dispro)n(v)n(e)27 b(the)i(conjecture)g (of)g(Esw)n(arathasan)e(and)i(Levine)f(on)h(the)g(\014niteness)g(of)g FB(J)2699 5429 y Fo(p)2738 5417 y Fs(:)40 b(to)29 b(ac)n(hiev)n(e)e (this)i(outcome)e(in)0 5517 y(form)n(ula)f(\()338 5517 y SDict begin H.S end 338 5517 a 0 TeXcolorgray Fs(1)p 0 TeXcolorgray 380 5463 a SDict begin H.R end 380 5463 a 380 5517 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.1) cvn H.B /ANN pdfmark end 380 5517 a Fs(\))j(the)g(sum)f(should)g (someho)n(w)e(v)-5 b(anish)28 b(in)g(v)-5 b(aluation.)38 b(W)-7 b(e)29 b(also)e(p)r(oin)n(t)h(out)h(that)g(alas)e(the)i (refutation)f(of)0 5616 y(case)e(\(2\))h(unfortunately)f(do)r(es)h(not) g(imply)d(Esw)n(arathasan)g(and)i(Levine's)g(conjecture,)h(or)f(ev)n (en)g(Bo)n(yd's)g(conjecture)0 5716 y(on)k(the)h(imp)r(ossibilit)n(y)25 b(of)31 b FB(\027)894 5728 y Fo(p)932 5716 y Fs(\()p FB(H)7 b Fs(\()p FB(n)p Fs(\)\))29 b Fq(\025)f Fs(4,)j(whic)n(h)e(w)n (ould)h(require)e(the)j(cases)f(\(1\))p FB(;)14 b Fs(\(2\))30 b(to)h(b)r(e)g(false,)f(and)h FB(M)36 b Fs(=)28 b(3)i(in)0 5816 y(case)d(\(3\).)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 10 10 TeXDict begin HPSdict begin 10 9 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -8 -191 a SDict begin H.S end -8 -191 a -8 -191 a SDict begin H.R end -8 -191 a -8 -191 a SDict begin [/View [/XYZ H.V]/Dest (page.10) cvn /DEST pdfmark end -8 -191 a 0 TeXcolorgray 0 -116 a Fp(10)700 b(LEONARDO)23 b(CAR)n(OFIGLIO,)g(LUIGI)f(DE)g(FILPO,)g(ALESSANDR)n(O)g(GAMBINI)p 0 TeXcolorgray 100 83 a Fs(One)30 b(p)r(ossible)e(w)n(a)n(y)h(to)i(use) f(the)h(results)e(ac)n(hiv)n(ed)f(could)h(b)r(e)i(that)g(one)f (undertak)n(en)g(b)n(y)g(Sanna)g([)3236 83 y SDict begin H.S end 3236 83 a 0 TeXcolorgray Fs(11)p 0 TeXcolorgray 3320 30 a SDict begin H.R end 3320 30 a 3320 83 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.sanna) cvn H.B /ANN pdfmark end 3320 83 a -1 w Fs(])h(to)f(\\coun)n(t")0 183 y(the)f(n)n(um)n(b)r(er)e(of)i(elemen)n(ts)e(of)h FB(J)1025 195 y Fo(p)1092 183 y Fs(with)h FB(p)p Fs(-adic)d(v)-5 b(aluation)26 b(greater)h(than)i(1.)40 b(One)28 b(could)g(try)g(to)h (coun)n(t)f(the)h(n)n(um)n(b)r(er)0 282 y(of)35 b(elemen)n(ts)d(with)j FB(p)p Fs(-adic)e(v)-5 b(aluation)32 b(greater)h(or)h(equal)f(to)i(a)f (p)r(ositiv)n(e)f(in)n(teger)g FB(k)k Fs(in)d(order)g(to)g(refute)h (case)f(2)h(of)0 382 y(Theorem)347 382 y SDict begin H.S end 347 382 a 0 TeXcolorgray Fs(2.5)p 0 TeXcolorgray 454 328 a SDict begin H.R end 454 328 a 454 382 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.2.5) cvn H.B /ANN pdfmark end 454 382 a Fs(.)g(The)24 b(problem)d(in)j(this)f(case)g(is)g (that)h(Sanna's)f(Lemma)f(2.2)h(on)h(short)f(in)n(terv)-5 b(als)21 b(w)n(orks)i(when)h(lo)r(oking)0 482 y(for)29 b(elemen)n(ts)e(that)i(cancel)f(the)i(deriv)-5 b(ativ)n(e)26 b(in)j(a)f(\014eld)h Fi(Z)1841 494 y Fo(p)1903 482 y Fs(\()p FB(f)1985 451 y Fk(0)1976 505 y Fo(d)2041 482 y Fq(\021)c Fs(0)84 b(mo)r(d)27 b FB(p)p Fs(\).)42 b(As)29 b(this)g(case)f(concerns)g(a)h(ring)f Fi(Z)3692 498 y Fo(p)3726 482 y Fm(k)0 588 y Fs(the)j(n)n(um)n(b)r(er)d(of)j(ro)r(ots)e (of)h(a)g(p)r(olynomial)25 b(cannot)30 b(b)r(e)g(b)r(ounded,)i(in)d (general,)f(b)n(y)i FB(p)p Fs(.)45 b(Bounding)28 b(it)i(b)n(y)g(\()p FB(d)21 b Fq(\000)e Fs(1\))p FB(p)3635 558 y Fo(k)q Fk(\000)p Fp(1)0 688 y Fs(leads)26 b(to)h(o)n(v)n(erestimating)c(the)28 b(ro)r(ots)f(and)g(then)h(to)g(a)f(c)n(hoice)e(of)j FB(z)j Fs(that)d(is)e(inadequate)g(for)h(our)g(purp)r(ose.)100 788 y(According)17 b(to)i(the)h(same)d(argumen)n(ts)g(used)j(in)e(this) h(pap)r(er)g(it)f(do)r(es)h(not)h(app)r(ear)e(to)r(o)h(di\016cult)f(to) h(extend)h(Theorems)0 887 y SDict begin H.S end 0 887 a 0 TeXcolorgray Fs(2.5)p 0 TeXcolorgray 107 834 a SDict begin H.R end 107 834 a 107 887 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.2.5) cvn H.B /ANN pdfmark end 107 887 a Fs(-)161 887 y SDict begin H.S end 161 887 a 0 TeXcolorgray Fs(2.6)p 0 TeXcolorgray 268 834 a SDict begin H.R end 268 834 a 268 887 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Theorem.2.6) cvn H.B /ANN pdfmark end 268 887 a 27 w Fs(for)27 b(the)h(generalized)c(harmonic)h(sum)1574 1084 y FB(H)1643 1096 y Fo(\013)1691 1084 y Fs(\()p FB(n)p Fs(\))e(:=)1979 980 y Fo(n)1939 1005 y Fl(X)1939 1184 y Fo(k)q Fp(=1)2109 1028 y Fs(1)p 2083 1065 94 4 v 2083 1141 a FB(k)2129 1117 y Fo(\013)0 1297 y Fs(in)i(the)g(case)g(\()p FB(p)14 b Fq(\000)f Fs(1\))23 b Fi(-)g FB(\013)p Fs(,)j(the)g(only)e (di\013erence)g(b)r(eing)h(that)g(the)h(descen)n(t)f(phenomenon)f(lo)n (w)n(ers)f(the)i(v)-5 b(aluation)23 b(b)n(y)i FB(\013)p Fs(.)0 1388 y SDict begin H.S end 0 1388 a 0 1388 a SDict begin 12 H.A end 0 1388 a 0 1388 a SDict begin [/View [/XYZ H.V]/Dest (section.6) cvn /DEST pdfmark end 0 1388 a 1438 1467 a Fs(6.)41 b Fr(A)n(ckno)n(wledgments)100 1616 y Fs(This)30 b(researc)n(h)f(is)h(the)h(result)f(of)h(a)f(collab)r (orativ)n(e)c(initiativ)n(e)h(among)i(studen)n(ts)i(of)g(the)h(three-y) n(ear)d(degree)h(pro-)0 1716 y(gramme)36 b(in)i(Mathematics)f(who)h(ha) n(v)n(e)g(activ)n(ely)e(participated)h(in)h(the)h(paths)g(of)g (excellence)d(promoted)i(b)n(y)g(the)0 1815 y(Departmen)n(t)27 b(of)g(Mathematics)e(\\Guido)i(Casteln)n(uo)n(v)n(o")c(of)28 b(Sapienza)e(Univ)n(ersit)n(y)f(of)i(Rome.)100 1915 y(W)-7 b(e)36 b(w)n(ould)e(also)g(lik)n(e)f(to)j(thank)g(Domenico)e(Marino,)h (Daniele)f(Saracino)f(and)j(F)-7 b(rancesco)34 b(V)-7 b(ertucci)35 b(for)h(their)0 2015 y(fruitful)27 b(suggestions)e(that)i (impro)n(v)n(ed)e(our)i(pap)r(er.)100 2114 y(W)-7 b(e)28 b(\014nally)d(w)n(ould)h(lik)n(e)f(to)j(thank)f(the)h(anon)n(ymous)d (referees)i(for)g(their)f(helpful)h(commen)n(ts)e(and)i(suggestions.) 1638 2284 y SDict begin H.S end 1638 2284 a 1638 2284 a SDict begin 12 H.A end 1638 2284 a 1638 2284 a SDict begin [/View [/XYZ H.V]/Dest (section*.1) cvn /DEST pdfmark end 1638 2284 a Fr(References)0 2333 y SDict begin H.S end 0 2333 a 0 2333 a SDict begin 10 H.A end 0 2333 a 0 2333 a SDict begin [/View [/XYZ H.V]/Dest (cite.Bettin) cvn /DEST pdfmark end 0 2333 a 0 TeXcolorgray 35 2416 a Fz([1])p 0 TeXcolorgray 35 w(S.)c(Bettin,)j(G.)d(Molteni,)j(C.)d (Sanna,)h Fd(Smal)t(l)j(values)f(of)g(signe)l(d)g(harmonic)h(sums)p Fz(,)d(C.)f(R.)g(Math.)h(Acad.)g(Sci.)g(P)n(aris)g Fc(356)p Fz(,)e(no.)i(11-12)145 2500 y(\(2018\),)h(1062{1074.)0 2516 y SDict begin H.S end 0 2516 a 0 2516 a SDict begin 10 H.A end 0 2516 a 0 2516 a SDict begin [/View [/XYZ H.V]/Dest (cite.BleicherE1975) 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