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minerva-icon--edit"></span> <span>Edit</span> </a> </li> </ul> </nav>  <div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Lucky_prime&redirect=no" class="mw-redirect" title="Lucky prime">Lucky prime</a>)</span></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"><script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><section class="mf-section-0" id="mf-section-0"> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about the mathematical concept. For other uses, see <a href="/wiki/Lucky_number_(disambiguation)" class="mw-disambig" title="Lucky number (disambiguation)">Lucky number (disambiguation)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Fortunate_number" title="Fortunate number">Fortunate number</a> or <a href="/wiki/Numerology" title="Numerology">Numerology</a>.</div> <p>In <a href="/wiki/Number_theory" title="Number theory">number theory</a>, a <b>lucky number</b> is a <a href="/wiki/Natural_number" title="Natural number">natural number</a> in a set which is generated by a certain "<a href="/wiki/Sieve_theory" title="Sieve theory">sieve</a>". This sieve is similar to the <a href="/wiki/Sieve_of_Eratosthenes" title="Sieve of Eratosthenes">sieve of Eratosthenes</a> that generates the <a href="/wiki/Prime_number" title="Prime number">primes</a>, but it eliminates numbers based on their position in the remaining set, instead of their value (or position in the initial set of natural numbers).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The term was introduced in 1956 in a paper by Gardiner, Lazarus, <a href="/wiki/Nicholas_Metropolis" title="Nicholas Metropolis">Metropolis</a> and <a href="/wiki/Stanislaw_Ulam" class="mw-redirect" title="Stanislaw Ulam">Ulam</a>. In the same work they also suggested calling another sieve, "the sieve of <a href="/wiki/Josephus" title="Josephus">Josephus</a> Flavius"<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> because of its similarity with the counting-out game in the <a href="/wiki/Josephus_problem" title="Josephus problem">Josephus problem</a>. </p><p>Lucky numbers share some properties with primes, such as asymptotic behaviour according to the <a href="/wiki/Prime_number_theorem" title="Prime number theorem">prime number theorem</a>; also, a version of <a href="/wiki/Goldbach%27s_conjecture" title="Goldbach's conjecture">Goldbach's conjecture</a> has been extended to them. There are infinitely many lucky numbers. Twin lucky numbers and <a href="/wiki/Twin_prime" title="Twin prime">twin primes</a> also appear to occur with similar frequency. However, if <i>L</i><sub><i>n</i></sub> denotes the <i>n</i>-th lucky number, and <i>p</i><sub><i>n</i></sub> the <i>n</i>-th prime, then <i>L</i><sub><i>n</i></sub> > <i>p</i><sub><i>n</i></sub> for all sufficiently large <i>n</i>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Because of their apparent similarities with the prime numbers, some mathematicians have suggested that some of their common properties may also be found in other sets of numbers generated by sieves of a certain unknown form, but there is little theoretical basis for this <a href="/wiki/Conjecture" title="Conjecture">conjecture</a>. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#The_sieving_process"><span class="tocnumber">1</span> <span class="toctext">The sieving process</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Lucky_primes"><span class="tocnumber">2</span> <span class="toctext">Lucky primes</span></a></li> <li class="toclevel-1 tocsection-3"><a href="#See_also"><span class="tocnumber">3</span> <span class="toctext">See also</span></a></li> <li class="toclevel-1 tocsection-4"><a href="#References"><span class="tocnumber">4</span> <span class="toctext">References</span></a></li> <li class="toclevel-1 tocsection-5"><a href="#Further_reading"><span class="tocnumber">5</span> <span class="toctext">Further reading</span></a></li> <li class="toclevel-1 tocsection-6"><a href="#External_links"><span class="tocnumber">6</span> <span class="toctext">External links</span></a></li> </ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="The_sieving_process">The sieving process</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Lucky_number&action=edit&section=1" title="Edit section: The sieving process" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-1 collapsible-block" id="mf-section-1"> <figure typeof="mw:File/Frame"><a href="/wiki/File:LuckySieve.gif" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/7/7c/LuckySieve.gif" decoding="async" width="299" height="364" class="mw-file-element" data-file-width="299" data-file-height="364"></noscript><span class="lazy-image-placeholder" style="width: 299px;height: 364px;" data-src="//upload.wikimedia.org/wikipedia/commons/7/7c/LuckySieve.gif" data-width="299" data-height="364" data-class="mw-file-element"> </span></a><figcaption>An animation demonstrating the lucky number sieve. The numbers on a reddish orange background are lucky numbers. When a number is eliminated its background changes from grey to purple. Chart goes to 120.</figcaption></figure> <table> <tbody><tr> <td colspan="25">Begin with a list of <a href="/wiki/Integer" title="Integer">integers</a> starting with 1: </td></tr> <tr> <td>1</td> <td>2</td> <td>3</td> <td>4</td> <td>5</td> <td>6</td> <td>7</td> <td>8</td> <td>9</td> <td>10</td> <td>11</td> <td>12</td> <td>13</td> <td>14</td> <td>15</td> <td>16</td> <td>17</td> <td>18</td> <td>19</td> <td>20</td> <td>21</td> <td>22</td> <td>23</td> <td>24</td> <td>25 </td></tr> <tr> <td colspan="25">Every second number (all <a href="/wiki/Even_number" class="mw-redirect" title="Even number">even numbers</a>) in the list is eliminated, leaving only the odd integers: </td></tr> <tr> <td>1</td> <td></td> <td>3</td> <td></td> <td>5</td> <td></td> <td>7</td> <td></td> <td>9</td> <td></td> <td>11</td> <td></td> <td>13</td> <td></td> <td>15</td> <td></td> <td>17</td> <td></td> <td>19</td> <td></td> <td>21</td> <td></td> <td>23</td> <td></td> <td>25 </td></tr> <tr> <td colspan="25">The first number remaining in the list after 1 is 3, so every third number (beginning at 1) which remains in the list (<i>not</i> every multiple of 3) is eliminated. The first of these is 5: </td></tr> <tr> <td>1</td> <td></td> <td>3</td> <td></td> <td></td> <td></td> <td>7</td> <td></td> <td>9</td> <td></td> <td></td> <td></td> <td>13</td> <td></td> <td>15</td> <td></td> <td></td> <td></td> <td>19</td> <td></td> <td>21</td> <td></td> <td></td> <td></td> <td>25 </td></tr> <tr> <td colspan="25">The next surviving number is now 7, so every seventh remaining number is eliminated. The first of these is 19: </td></tr> <tr> <td>1</td> <td></td> <td>3</td> <td></td> <td></td> <td></td> <td>7</td> <td></td> <td>9</td> <td></td> <td></td> <td></td> <td>13</td> <td></td> <td>15</td> <td></td> <td></td> <td></td> <td></td> <td></td> <td>21</td> <td></td> <td></td> <td></td> <td>25 </td></tr></tbody></table> <p>Continue removing the <i>n</i>th remaining numbers, where <i>n</i> is the next number in the list after the last surviving number. Next in this example is 9. </p><p>One way that the application of the procedure differs from that of the Sieve of Eratosthenes is that for <i>n</i> being the number being multiplied on a specific pass, the first number eliminated on the pass is the <i>n</i>-th remaining number that has not yet been eliminated, as opposed to the number <i>2n</i>. That is to say, the list of numbers this sieve counts through is different on each pass (for example 1, 3, 7, 9, 13, 15, 19... on the third pass), whereas in the Sieve of Eratosthenes, the sieve always counts through the entire original list (1, 2, 3...). </p><p>When this procedure has been carried out completely, the remaining integers are the lucky numbers (those that happen to be prime are in bold): </p> <dl><dd><a href="/wiki/1_(number)" class="mw-redirect" title="1 (number)">1</a>, <b><a href="/wiki/3_(number)" class="mw-redirect" title="3 (number)">3</a></b>, <b><a href="/wiki/7_(number)" class="mw-redirect" title="7 (number)">7</a></b>, <a href="/wiki/9_(number)" class="mw-redirect" title="9 (number)">9</a>, <b><a href="/wiki/13_(number)" title="13 (number)">13</a></b>, <a href="/wiki/15_(number)" title="15 (number)">15</a>, <a href="/wiki/21_(number)" title="21 (number)">21</a>, <a href="/wiki/25_(number)" title="25 (number)">25</a>, <b><a href="/wiki/31_(number)" title="31 (number)">31</a></b>, <a href="/wiki/33_(number)" title="33 (number)">33</a>, <b><a href="/wiki/37_(number)" title="37 (number)">37</a></b>, <b><a href="/wiki/43_(number)" title="43 (number)">43</a></b>, <a href="/wiki/49_(number)" title="49 (number)">49</a>, <a href="/wiki/51_(number)" title="51 (number)">51</a>, <a href="/wiki/63_(number)" title="63 (number)">63</a>, <b><a href="/wiki/67_(number)" title="67 (number)">67</a></b>, <a href="/wiki/69_(number)" title="69 (number)">69</a>, <b><a href="/wiki/73_(number)" title="73 (number)">73</a></b>, <a href="/wiki/75_(number)" title="75 (number)">75</a>, <b><a href="/wiki/79_(number)" title="79 (number)">79</a></b>, <a href="/wiki/87_(number)" title="87 (number)">87</a>, <a href="/wiki/93_(number)" title="93 (number)">93</a>, <a href="/wiki/99_(number)" title="99 (number)">99</a>, <a href="/wiki/105_(number)" title="105 (number)">105</a>, <a href="/wiki/111_(number)" title="111 (number)">111</a>, <a href="/wiki/115_(number)" title="115 (number)">115</a>, <b><a href="/wiki/127_(number)" title="127 (number)">127</a></b>, <a href="/wiki/129_(number)" title="129 (number)">129</a>, <a href="/wiki/133_(number)" title="133 (number)">133</a>, <a href="/wiki/135_(number)" title="135 (number)">135</a>, <a href="/wiki/141_(number)" title="141 (number)">141</a>, <b><a href="/wiki/151_(number)" title="151 (number)">151</a></b>, <a href="/wiki/159_(number)" title="159 (number)">159</a>, <b><a href="/wiki/163_(number)" title="163 (number)">163</a></b>, <a href="/wiki/169_(number)" title="169 (number)">169</a>, <a href="/wiki/171_(number)" title="171 (number)">171</a>, <a href="/wiki/189_(number)" title="189 (number)">189</a>, <b><a href="/wiki/193_(number)" title="193 (number)">193</a></b>, <a href="/wiki/195_(number)" title="195 (number)">195</a>, <a href="/wiki/201_(number)" title="201 (number)">201</a>, <a href="/wiki/205_(number)" title="205 (number)">205</a>, <b><a href="/wiki/211_(number)" title="211 (number)">211</a></b>, <a href="/wiki/219_(number)" title="219 (number)">219</a>, <b><a href="/wiki/223_(number)" title="223 (number)">223</a></b>, <a href="/wiki/231_(number)" title="231 (number)">231</a>, <a href="/wiki/235_(number)" title="235 (number)">235</a>, <a href="/wiki/237_(number)" title="237 (number)">237</a>, <b><a href="/wiki/241_(number)" title="241 (number)">241</a></b>, <a href="/wiki/259_(number)" title="259 (number)">259</a>, <a href="/wiki/261_(number)" title="261 (number)">261</a>, <a href="/wiki/267_(number)" title="267 (number)">267</a>, <a href="/wiki/273_(number)" title="273 (number)">273</a>, <b><a href="/wiki/283_(number)" title="283 (number)">283</a></b>, <a href="/wiki/285_(number)" title="285 (number)">285</a>, <a href="/wiki/289_(number)" title="289 (number)">289</a>, <a href="/wiki/297_(number)" title="297 (number)">297</a>, <a href="/wiki/303_(number)" title="303 (number)">303</a>, <b><a href="/wiki/307_(number)" title="307 (number)">307</a></b>, <a href="/wiki/319_(number)" class="mw-redirect" title="319 (number)">319</a>, <a href="/wiki/321_(number)" class="mw-redirect" title="321 (number)">321</a>, <a href="/wiki/327_(number)" class="mw-redirect" title="327 (number)">327</a>, <b><a href="/wiki/331_(number)" class="mw-redirect" title="331 (number)">331</a></b>, <a href="/wiki/339_(number)" class="mw-redirect" title="339 (number)">339</a>, ... (sequence <span class="nowrap external"><a href="//oeis.org/A000959" class="extiw" title="oeis:A000959">A000959</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</dd></dl> <p>The lucky number which removes <i>n</i> from the list of lucky numbers is: (0 if <i>n</i> is a lucky number) </p> <dl><dd>0, 2, 0, 2, 3, 2, 0, 2, 0, 2, 3, 2, 0, 2, 0, 2, 3, 2, 7, 2, 0, 2, 3, 2, 0, 2, 9, 2, 3, 2, 0, 2, 0, 2, 3, 2, 0, 2, 7, 2, 3, 2, 0, 2, 13, 2, 3, 2, 0, 2, 0, 2, 3, 2, 15, 2, 9, 2, 3, 2, 7, 2, 0, 2, 3, 2, 0, 2, 0, 2, 3, 2, 0, 2, 0, 2, 3, 2, 0, 2, 7, 2, 3, 2, 21, 2, ... (sequence <span class="nowrap external"><a href="//oeis.org/A264940" class="extiw" title="oeis:A264940">A264940</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Lucky_primes">Lucky primes</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Lucky_number&action=edit&section=2" title="Edit section: Lucky primes" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-2 collapsible-block" id="mf-section-2"> <p>A "lucky prime" is a lucky number that is prime. They are: </p> <dl><dd>3, 7, 13, 31, 37, 43, 67, 73, 79, 127, 151, 163, 193, 211, 223, 241, 283, 307, 331, 349, 367, 409, 421, 433, 463, 487, 541, 577, 601, 613, 619, 631, 643, 673, 727, 739, 769, 787, 823, 883, 937, 991, 997, ... (sequence <span class="nowrap external"><a href="//oeis.org/A031157" class="extiw" title="oeis:A031157">A031157</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</dd></dl> <p>It has been conjectured that there are infinitely many lucky primes.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="See_also">See also</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Lucky_number&action=edit&section=3" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-3 collapsible-block" id="mf-section-3"> <ul><li><a href="/wiki/Lucky_numbers_of_Euler" title="Lucky numbers of Euler">Lucky numbers of Euler</a></li> <li><a href="/wiki/Fortunate_number" title="Fortunate number">Fortunate number</a></li> <li><a href="/wiki/Happy_number" title="Happy number">Happy number</a></li> <li><a href="/wiki/Harshad_number" title="Harshad number">Harshad number</a></li> <li><a href="/wiki/Josephus_problem" title="Josephus problem">Josephus problem</a></li> <li><a href="/wiki/Gambling" title="Gambling">Gambling</a></li> <li><a href="/wiki/Lottery" title="Lottery">Lottery</a></li> <li><a href="/wiki/Keno" title="Keno">Keno</a></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="References">References</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Lucky_number&action=edit&section=4" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-4 collapsible-block" id="mf-section-4"> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFWeisstein,_Eric_W." class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/LuckyNumber.html">"Lucky Number"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Lucky+Number&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FLuckyNumber.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALucky+number" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGardinerLazarusMetropolisUlam1956" class="citation journal cs1">Gardiner, Verna; Lazarus, R.; <a href="/wiki/Nicholas_Metropolis" title="Nicholas Metropolis">Metropolis, N.</a>; <a href="/wiki/Stanislaw_Ulam" class="mw-redirect" title="Stanislaw Ulam">Ulam, S.</a> (1956). "On certain sequences of integers defined by sieves". <i><a href="/wiki/Mathematics_Magazine" title="Mathematics Magazine">Mathematics Magazine</a></i>. <b>29</b> (3): 117–122. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3029719">10.2307/3029719</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0025-570X">0025-570X</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3029719">3029719</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0071.27002">0071.27002</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=On+certain+sequences+of+integers+defined+by+sieves&rft.volume=29&rft.issue=3&rft.pages=117-122&rft.date=1956&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0071.27002%23id-name%3DZbl&rft.issn=0025-570X&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3029719%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F3029719&rft.aulast=Gardiner&rft.aufirst=Verna&rft.au=Lazarus%2C+R.&rft.au=Metropolis%2C+N.&rft.au=Ulam%2C+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALucky+number" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHawkinsBriggs1957" class="citation journal cs1">Hawkins, D.; Briggs, W.E. (1957). "The lucky number theorem". <i><a href="/wiki/Mathematics_Magazine" title="Mathematics Magazine">Mathematics Magazine</a></i>. <b>31</b> (2): 81–84, 277–280. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3029213">10.2307/3029213</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0025-570X">0025-570X</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3029213">3029213</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0084.04202">0084.04202</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=The+lucky+number+theorem&rft.volume=31&rft.issue=2&rft.pages=81-84%2C+277-280&rft.date=1957&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0084.04202%23id-name%3DZbl&rft.issn=0025-570X&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3029213%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F3029213&rft.aulast=Hawkins&rft.aufirst=D.&rft.au=Briggs%2C+W.E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALucky+number" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id='CITEREFSloane_"A031157"' class="citation web cs1"><a href="/wiki/Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.). <a rel="nofollow" class="external text" href="https://oeis.org/A031157">"Sequence A031157 (Numbers that are both lucky and prime)"</a>. <i>The <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a></i>. OEIS Foundation.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+On-Line+Encyclopedia+of+Integer+Sequences&rft.atitle=Sequence%26%23x20%3BA031157%26%23x20%3B%28Numbers+that+are+both+lucky+and+prime%29&rft_id=https%3A%2F%2Foeis.org%2FA031157&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALucky+number" class="Z3988"></span></span> </li> </ol></div></div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Lucky_number&action=edit&section=5" title="Edit section: Further reading" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-5 collapsible-block" id="mf-section-5"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuy2004" class="citation book cs1"><a href="/wiki/Richard_K._Guy" title="Richard K. Guy">Guy, Richard K.</a> (2004). <i>Unsolved problems in number theory</i> (3rd ed.). <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. C3. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-20860-2" title="Special:BookSources/978-0-387-20860-2"><bdi>978-0-387-20860-2</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1058.11001">1058.11001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Unsolved+problems+in+number+theory&rft.pages=C3&rft.edition=3rd&rft.pub=Springer-Verlag&rft.date=2004&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1058.11001%23id-name%3DZbl&rft.isbn=978-0-387-20860-2&rft.aulast=Guy&rft.aufirst=Richard+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALucky+number" class="Z3988"></span></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="External_links">External links</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Lucky_number&action=edit&section=6" title="Edit section: External links" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-6 collapsible-block" id="mf-section-6"> <ul><li><a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/LuckyNumbers/">Lucky Numbers</a> by Enrique Zeleny, <a href="/wiki/The_Wolfram_Demonstrations_Project" class="mw-redirect" title="The Wolfram Demonstrations Project">The Wolfram Demonstrations Project</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSymonds" class="citation web cs1">Symonds, Ria. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160919165741/http://www.numberphile.com/videos/lucky_numbers.html">"31: And other lucky numbers"</a>. <i>Numberphile</i>. <a href="/wiki/Brady_Haran" title="Brady Haran">Brady Haran</a>. Archived from <a rel="nofollow" class="external text" href="http://www.numberphile.com/videos/lucky_numbers.html">the original</a> on 2016-09-19<span class="reference-accessdate">. Retrieved <span class="nowrap">2013-04-02</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Numberphile&rft.atitle=31%3A+And+other+lucky+numbers&rft.aulast=Symonds&rft.aufirst=Ria&rft_id=http%3A%2F%2Fwww.numberphile.com%2Fvideos%2Flucky_numbers.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALucky+number" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist 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href="https://zh-yue.wikipedia.org/wiki/%E5%B9%B8%E9%81%8B%E6%95%B8" title="幸運數 – Cantonese" lang="yue" hreflang="yue" data-title="幸運數" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%B9%B8%E8%BF%90%E6%95%B0" title="幸运数 – Chinese" lang="zh" hreflang="zh" data-title="幸运数" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li></ul> </section> </div> <div class="minerva-footer-logo"><img src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" alt="Wikipedia" width="120" height="18" style="width: 7.5em; height: 1.125em;"/> </div> <ul id="footer-info" class="footer-info hlist hlist-separated"> <li id="footer-info-lastmod"> This page was last edited on 27 September 2024, at 03:30<span class="anonymous-show"> (UTC)</span>.</li> <li 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