Editing Sierpiński number (section)

==Sierpiński problem==

{{unsolved|mathematics|Is 78,557 the smallest Sierpiński number?}}

The '''Sierpiński problem''' asks for the value of the smallest Sierpiński number. In private correspondence with [[Paul Erdős]], Selfridge [[conjecture]]d that 78,557 was the smallest Sierpiński number.<ref>{{Cite journal|last1=Erdős|first1=Paul|author1-link=Paul Erdős|last2=Odlyzko|first2=Andrew Michael|author2-link=Andrew Odlyzko|date=May 1, 1979|title=On the density of odd integers of the form {{math|(''p'' − 1)2<sup>−''n''</sup>}} and related questions|journal=[[Journal of Number Theory]]|publisher=[[Elsevier]]|volume=11|issue=2|page=258|language=en|doi=10.1016/0022-314X(79)90043-X|issn=0022-314X|doi-access=free}}</ref> No smaller Sierpiński numbers have been discovered, and it is now believed that 78,557 is the smallest number.<ref>{{Cite book|last1=Guy|first1=Richard Kenneth|author1-link=Richard K. Guy|year=2005|title=Unsolved Problems in Number Theory|publisher=[[Springer Science+Business Media|Springer-Verlag]]|location=New York|language=en|isbn=978-0-387-20860-2|pages=B21:119{{ndash}}121, F13:383{{ndash}}385|oclc=634701581}}</ref>

To show that 78,557 really is the smallest Sierpiński number, one must show that all the odd numbers smaller than 78,557 are ''not'' Sierpiński numbers. That is, for every odd ''k'' below 78,557, there needs to exist a positive integer ''n'' such that {{math|''k''2<sup>''n''</sup> + 1}} is prime.<ref name="PG" /> The distributed volunteer computing project [[PrimeGrid]] is attempting to eliminate all the remaining values of ''k'':<ref>{{Cite web|url=https://www.primegrid.com/stats_sob_llr.php|title=Seventeen or Bust statistics|website=PrimeGrid|access-date=November 21, 2019}}</ref>

: ''k'' = 21181, 22699, 24737, 55459, and 67607.