Does listing the prime factors always stop? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T16:33:15Z http://mathoverflow.net/feeds/question/31406 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/31406/does-listing-the-prime-factors-always-stop Does listing the prime factors always stop? Franklin 2010-07-11T13:10:02Z 2010-07-11T14:20:52Z <p>Take a natural number's prime factors and list them increasingly and repeating them according to multiplicity. Concatenate their decimal (or in any base) representation to get a new number and repeat the process. Does this always end in a prime number for any input? </p> http://mathoverflow.net/questions/31406/does-listing-the-prime-factors-always-stop/31409#31409 Answer by Wadim Zudilin for Does listing the prime factors always stop? Wadim Zudilin 2010-07-11T14:02:30Z 2010-07-11T14:02:30Z <p>Is too long for comment: \$\ddot\smile\$</p> <pre><code>77 711 3379 31109 132393 344131 1731653 71143523 11115771019 31135742029 717261644891 11193431873899 116134799345907 3204751189066719 31068250396355573 62161149980213343 336906794442245927 734615161567701999 31318836286194043641 333431436916146111627309 33205716184556772142207827 31367222155734752971376323127 733915126325777821480557336017 476734743112036198712947236602187 377171280957470909577133234490256751 3096049809383121823389214993262890297 73796236325118712936424989555929478399 13118114526141133089538087518197265265053 319521441731977174163487542111577539726749 595415617656474189392601483764603009147911 13842314669573706744784027901056001426046777 3129192501509379967095393172011476342474406759 3203927133121399320591151296378525102203388346189 133119651853912195249113288820301002347322382772769 11103725795898241052711667094407302642807490159301277 1152194718705941109372661574127837007959097317735411121 6318653972357749718234812726673333988788742328093848793 711111311391974493533533521186754240313734089696843349346661 3771113711016948131790459407678947892694155341923379077407684653 7310113562312583178332057129971031882457634609852680847686251943317 3111197172271564982895268105721087453190074064393495190773755017652247 373111539295698434141591345095168649790005875768086611455076505611166279 33333711151101316117103176926136887884135060403955118931001222053567659972075047 37987951744462008749649348751784002342702203325604103216176784227054268232116293 ................ </code></pre> <p>Although in this specific example there is still a chance to arrive at a prime number, the heuristic is against this conclusion as the length of record increases while prime numbers appear "rarer".</p> http://mathoverflow.net/questions/31406/does-listing-the-prime-factors-always-stop/31411#31411 Answer by Nurdin Takenov for Does listing the prime factors always stop? Nurdin Takenov 2010-07-11T14:15:11Z 2010-07-11T14:15:11Z <p>It's open problem, sequence <a href="http://www.research.att.com/~njas/sequences/A037274" rel="nofollow">A037274</a> from OEIS, so-called "<a href="http://mathworld.wolfram.com/HomePrime.html" rel="nofollow">home primes</a>". Hm, the value for n=77 is even unknown.</p> <p>P.S. <a href="http://www.research.att.com/~njas/sequences/" rel="nofollow">On-Line Encyclopedia of Integer Sequences</a> definitely should be included in FAQ.</p>