Date: Tue, 9 Mar 2004 01:08:57 -0600 (CST) From: Richard McIntosh <mcintosh@math.uregina.ca> To: Paul dot Zimmermann at loria dot fr X-UIDL: &Yh!!07+!!kP("!iA&"! Here is the latest update on the Wieferich, Wilson, Wall-Sun-Sun (Fibonacci Wieferich) and Wolstenholme search. The Wieferich search is complete to 1.25 * 10^15, the Wall-Sun-Sun to 100 * 10^12, the Wilson to 500 * 10^6 and the Wolstenholme to 10^9. Best Regards, Richard McIntosh ============================================================== WIEFERICH STATUS (June 2001) 0-10billion: DONE by KD (RX = "rigor-check") 10-30billion: DONE by KD - RX redone 30-40billion: DONE by REC - RX redone 40-50billion: DONE by KD - RX 50-90billion: DONE by REC - RX } > 50-100bill 90-200billion: DONE by KD - RX } redone 0-200billion: checked by RM, A-table agrees 200-304billion: DONE by REC - RX 203-279billion: Pentium RETRY DONE by KD - RX 291-292billion: Pentium RETRY DONE by KD - RX 400-425billion: Pentium RETRY DONE by KD - RX 304-399billion: DONE by KD - RX 399-431billion: DONE by REC - RX 431-900billion: DONE by KD - RX 900-905billion: DONE by TEC - RX 905bill-1trillion: DONE by KD(i) - RX 1-1.12trillion: DONE by KD(i) - RX 1.12-1.2trillion: DONE by KD - RX 1.2-1.26trillion: DONE by KD(i) - RX 1.26-1.7trillion: DONE by KD - RX 1.7-2trillion: DONE by REC - RX 2-2.1trillion: DONE by RM - RX 0-2trillion: checked by DB, A-table agrees 2-4trillion: DONE by DB 4-8trillion: DONE by RM 8-49trillion: DONE by RB 49-200trillion: DONE by JC 200-1250trillion: DONE by JK, JR & others Prime p is a Wieferich prime if 2^(p-1) = 1 (mod p^2). The only known Wieferich primes are 1093 and 3511. Table of special instances (|A| <= 100) of 2^((p-1)/2) (mod p^2); p > 1 billion. p +-1 + A p 1222336487 +1 + 60 p 1259662487 +1 - 71 p 1274153897 +1 - 86 p 1494408397 -1 + 52 p 1584392531 -1 - 24 p 1586651309 -1 - 24 p 1662410923 -1 - 70 p 1817972423 +1 - 56 p 1890830857 +1 + 69 p 2062661389 -1 + 55 p 2244893621 -1 + 47 p 2332252547 -1 - 33 p 2416644757 -1 + 67 p 2461090421 -1 + 47 p 2566816313 +1 + 52 p 2570948153 +1 - 41 p 2589186937 +1 - 85 p 2709711233 +1 + 50 p 2760945133 -1 - 77 p 2954547209 +1 + 32 p 3027263587 -1 - 95 p 3133652447 +1 + 87 p 3303616961 +1 - 20 p 3520624567 +1 - 6 p 3606693551 +1 + 21 p 4449676157 -1 - 15 p 5045920247 +1 + 76 p 5409537149 -1 + 66 p 8843450093 -1 - 20 p 10048450537 +1 + 82 p 10329891503 +1 + 79 p 11214704947 -1 + 56 p 20051397221 -1 - 46 p 20366156849 +1 - 95 p 36673326289 +1 - 45 p 46262476201 +1 + 5 p 47004625957 -1 + 1 p 49819566449 +1 + 27 p 53359191887 +1 + 50 p 58481216789 -1 + 5 p 76843523891 -1 + 1 p 82834772291 -1 + 82 p 108058158839 +1 + 58 p 130861186019 -1 - 97 p 138528575509 -1 - 23 p 239398882511 +1 - 72 p 252137567497 +1 - 31 p 252074060191 +1 + 61 p 299948374351 +1 - 19 p 405897532891 -1 + 61 p 443168739911 +1 + 64 p 504568016327 +1 + 55 p 703781283787 -1 - 49 p 955840782881 +1 - 84 p 980377925057 +1 - 90 p 981086885117 -1 + 85 p 1095406033573 -1 + 42 p 1104406423781 -1 + 54 p 1180032105761 +1 - 6 p 1722721869859 -1 - 31 p 1730418792409 +1 - 46 p 1780536689159 +1 + 84 p 2207775149407 +1 - 99 p 2424653846701 -1 - 51 p 2610372685663 +1 - 28 p 3667691800441 +1 + 27 p 3713054321579 -1 - 51 p 3729819224423 +1 + 77 p 4006528141163 -1 + 17 p 4169357937293 -1 - 27 p 5216344035949 -1 + 93 p 5240305919047 +1 - 95 p 7355288787229 -1 - 68 p 7876427903107 -1 - 48 p 8851776421399 +1 - 81 p 11344191252809 +1 - 92 p 12456646902457 +1 + 2 p 15056776355693 -1 - 19 p 23639424831877 -1 - 48 p 24990087401551 +1 + 16 p 28506780213511 +1 - 28 p 28785529445977 +1 + 33 p 29230410915073 +1 + 96 p 30189412701163 -1 - 37 p 30309769394167 +1 + 28 p 63735899194511 +1 - 16 p 63918629031731 -1 + 38 p 67961346537659 -1 - 49 p 68132247624521 +1 - 55 p 92226580839683 -1 - 76 p 118485210646981 -1 - 90 p 134257821895921 +1 + 10 p 153332502585091 -1 + 59 p 181841793213263 +1 + 90 p 205250817470827 -1 - 78 p 259990715684839 +1 - 12 p 339258218134349 -1 + 2 p 342092449620191 +1 + 90 p 346412396858131 -1 - 48 p 362061154308767 +1 - 64 p 694936752678643 -1 + 75 p 696740841781447 +1 + 61 p 734180764265903 +1 + 37 p 739507312099561 +1 - 78 p 765760560131939 -1 + 38 p 1140417231387373 -1 - 82 p 1170553064286511 +1 - 84 p ============================================================== WILSON STATUS (Feb. 1999): 0-20mill DONE by KD - RX 20-51mill DONE by REC - RX 51-54mill DONE by KD - RX 54-200mill DONE by REC - RX($) 0-100mill checked by RM ($) CAUTION: The 1-mill intervals beginning with the following were resolved using (2.5) of the CDP Draft of 14 Jan 1995. 193-200, 189, 131-132, 136, 145, 147, 150, 152, 165-166,170-172, 175-186 200-350mill DONE by REC - RX($) 350-400mill DONE by KD+REC - RX($) 400-410mill DONE by REC - RX($) 410-500mill DONE by REC Prime p is a Wilson prime if (p-1)! = -1 (mod p^2). The only known Wilson primes are 5, 13 and 563. Table of special instances (|B| <= 100) of (p-1)! (mod p^2); 1 million < p < 100 million. p -1 + B p 1282279 -1 + 20 p 1306817 -1 - 30 p 1308491 -1 - 55 p 1433813 -1 - 32 p 1638347 -1 - 45 p 1640147 -1 - 88 p 1647931 -1 + 14 p 1666403 -1 + 99 p 1750901 -1 + 34 p 1851953 -1 - 50 p 2031053 -1 - 18 p 2278343 -1 + 21 p 2313083 -1 + 15 p 2695933 -1 - 73 p 3640753 -1 + 69 p 3677071 -1 - 32 p 3764437 -1 - 99 p 3958621 -1 + 75 p 5062469 -1 + 39 p 5063803 -1 + 40 p 6331519 -1 + 91 p 6706067 -1 + 45 p 7392257 -1 + 40 p 8315831 -1 + 3 p 8871167 -1 - 85 p 9278443 -1 - 75 p 9615329 -1 + 27 p 9756727 -1 + 23 p 10746881 -1 - 7 p 11465149 -1 - 62 p 11512541 -1 - 26 p 11892977 -1 - 7 p 12632117 -1 - 27 p 12893203 -1 - 53 p 14296621 -1 + 2 p 16711069 -1 + 95 p 16738091 -1 + 58 p 17879887 -1 + 63 p 19344553 -1 - 93 p 19365641 -1 + 75 p 20951477 -1 + 25 p 20972977 -1 + 58 p 21561013 -1 - 90 p 23818681 -1 + 23 p 27783521 -1 - 51 p 27812887 -1 + 21 p 29085907 -1 + 9 p 29327513 -1 + 13 p 30959321 -1 + 24 p 33187157 -1 + 60 p 33968041 -1 + 12 p 39198017 -1 - 7 p 45920923 -1 - 63 p 51802061 -1 + 4 p 53188379 -1 - 54 p 56151923 -1 - 1 p 57526411 -1 - 66 p 64197799 -1 + 13 p 72818227 -1 - 27 p 87467099 -1 - 2 p 91926437 -1 - 32 p 92191909 -1 + 94 p 93445061 -1 - 30 p 93559087 -1 - 3 p 94510219 -1 - 69 p Table of special instances (-100 <= B <= 0) of (p-1)! (mod p^2); 100 million < p < 500 million. p -1 + B p 101710369 -1 - 70 p 117385529 -1 - 43 p 212911781 -1 - 92 p 216331463 -1 - 36 p 327357841 -1 - 62 p 411237857 -1 - 84 p 479163953 -1 - 50 p Table of large factorials: 2^36 + 117 -1 - 28030812110 p (Verified by RM) 2^40 + 625 -1 - 533091778023 p (Verified by RM) ============================================================== WILSON COMPOSITE STATUS (4 Aug 95 from KD) 0-155tho: DONE by KD VERIFIED by REC -RX 155-500tho: DONE by KD -RX 500-651tho: DONE by REC -RX 651-1511tho: DONE by KD -RX 1511-2000tho: DONE by KD -RX 2-10 mill: DONE by KD -RX Table of Wilson composites: 5971 = 7 * 853 (Kloss, 1965) 558771 = 3 * 19 * 9803 1964215 = 5 * 11 * 71 * 503 8121909 = 3 * 139 * 19477 ============================================================== WALL-SUN-SUN STATUS (Oct 2003) 0-60trill: DONE by RM 60-90trill: DONE by JK 86-100trill: DONE by RM Let F_n be the nth Fibonacci number. F_0 = 0, F_1 = 1, F_n = F_{n-1} + F_{n-2}. Let e = (p/5) be the Legendre symbol. e = 1 if p = +-1 (mod 5) and e = -1 if p = +-2 (mod 5). A prime p is a Wall-Sun-Sun prime if F_{p-e} = 0 (mod p^2). No Wall-Sun-Sun primes are known. Let L_n be the nth Lucas number. L_0 = 2, L_1 = 1, L_n = L_{n-1} + L_{n-2}. L_{p-e} = 2e (mod p^2) for all primes p > 5. Theorem: Let p be prime. Then the following are equivalent (i) F_{p-e} = 0 (mod p^2), (ii) F_p = e (mod p^2), (iii) L_p = 1 (mod p^2). Table of special instances of (|C| <= 100) of F_(p-(p/5)) (mod p^2); p > 1 billion. p 0 + C p 1041968177 -3 p 1132487479 10 p 1394046523 89 p 1529242499 32 p 1722432611 -21 p 1734471553 -37 p 1859934553 74 p 2025051311 -4 p 2387160103 -78 p 2566493471 81 p 3053448691 -86 p 3340099687 -29 p 3479663357 -45 p 3532361671 26 p 6627557731 -43 p 6741860329 80 p 6859114489 55 p 8352762221 94 p 9760159421 -30 p 10115341939 70 p 10536116749 66 p 10612008943 -69 p 11002117921 95 p 11025166637 23 p 12216759923 -33 p 12387724249 39 p 13585864301 57 p 13699540891 73 p 13941800291 15 p 16368086681 50 p 16548311011 92 p 17370126353 -70 p 18526398173 -29 p 20488487861 -92 p 24178397183 3 p 25049632411 51 p 31811337589 -89 p 37883499127 -10 p 50261755937 -73 p 65543096747 -36 p 74176637257 -70 p 82202291813 -11 p 87918267869 78 p 89845144679 51 p 94903475011 -15 p 101876918491 87 p 110717352637 -77 p 111646394549 -86 p 115301883659 60 p 115364673283 62 p 129316722167 -22 p 134431860461 -30 p 166466703223 64 p 170273590301 78 p 233642484991 89 p 277764184829 64 p 283750593739 37 p 300258464153 70 p 334015396151 79 p 442650398821 74 p 458432241569 -61 p 621291852133 96 p 762383958397 -86 p 766193665711 -8 p 800537116979 -20 p 1082150673011 -57 p 1171853196853 -30 p 1551559563569 4 p 1786416720937 -82 p 1996100161327 98 p 2669682790919 -10 p 3311519272973 -47 p 3814438808399 -56 p 3858738583171 44 p 5457214172491 76 p 8030311150847 -12 p 10591568377751 33 p 11406840440243 26 p 11456600879363 -41 p 12801531958729 17 p 13860200708287 -5 p 18801391545961 33 p 21960966892313 54 p 22548139284371 -16 p 25186595067349 -59 p 26861987497291 -28 p 28263796914043 -88 p 39568597208207 53 p 46006966741789 -59 p 64241561031937 -44 p 67721845179979 -52 p 75320741942123 71 p 82789107950701 -42 p 85136199318719 -92 p 86040142362653 19 p 88536418898561 99 p 90299689540433 56 p 91486955300761 -71 p ================================================================= WOLSTENHOLME STATUS by RM (March 2004): A prime p is a Wolstenholme prime if the central binomial coefficient (2p choose p) = 2 (mod p^4), or equivalently, B_{p-3} = 0 (mod p), where B_n is the nth Bernoulli number. A prime p > 7 is a Wolstenholme prime if and only if sum(1/k^3, k = [p/6] + 1, ..., [p/4]) = 0 (mod p). The only known Wolstenholme primes are 16843 and 2124679. The search is complete up to 10^9. ================================================================= SOME LARGE PROBABLE-PRIMES by RM: (2^42737 + 1)/3, (2^83339 + 1)/3 and (2^95369 + 1)/3. (2^(4p) + 1)/17 is prime for p = 3, 5, 7, 23, 37, 89, 149, 173, 251, 307, 317 and no other values of p < 30197. It is a probable-prime for p = 30197. =================================================================