Wacław Franciszek Sierpiński (14 March 1882 — 21 October 1969), a Polish mathematician, was known for outstanding contributions to set theory, number theory, theory of functions and topology. It is in number theory where we find the Sierpinski problem.
Basically, the Sierpinski problem is "What is the smallest Sierpinski number" and the prime Sierpinski problem is "What is the smallest prime Sierpinski number?"
First we look at Proth numbers (named after the French mathematician François Proth). A Proth number is a number of the form k*2^n+1 where k is odd, n is a positive integer, and 2^n>k.
A Sierpinski number is an odd k such that the Proth number k*2^n+1 is not prime for all n. For example, 3 is not a Sierpinski number because n=2 produces a prime number (3*2^2+1=13). In 1962, John Selfridge proved that 78,557 is a Sierpinski number, meaning he showed that for all n, 78557*2^n+1 was not prime.
Most number theorists believe that 78,557 is the smallest Sierpinski number, but it hasn\'t yet been proven. In order to prove it, it has to be shown that every single k less than 78,557 is not a Sierpinski number, and to do that, some n must be found that makes k*2^n+1 prime.
The smallest proven prime Sierpinski number is 271,129. In order to prove it, it has to be shown that every single prime k less than 271,129 is not a Sierpinski number, and to do that, some n must be found that makes k*2^n+1 prime.
Seventeen or Bust is working on the Sierpinski problem and the Prime Sierpinski Project is working on the prime Sierpinski problem. The following k's remain for each project
- 21181, 22699, 24737, 55459, 67607
'prime' Sierpinski problem
- 22699*, 67607*, 79309, 79817, 152267, 156511, 222113, 225931, 237019,
- * means being tested by Seventeen or Bust
For more information about PSP, please see:
For more information about Sierpinski, Sierpinski number, and the Sierpinsk problem, please see these resources:
- Wacław Franciszek Sierpiński (WIKI)
- Sierpinski number / The Sierpinski problem (WIKI)
- The Sierpinski problem (Prothsearch)
258317*2^5450519+1 is prime! (found by Sloth@PSP on 28/7/2008)
90527*2^9162167+1 is prime! (found by Bold_Seeker@PSP on 19/6/2010)
168451*2^19375200+1 is prime! Found by Ben Maloney (paleseptember) on September 17th, 2017. (official announcement)