About the Prime Sierpinski Problem

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 that it is the smallest Sierpinski number, 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 that it is the smallest prime Sierpinski number, 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.

Previously, PrimeGrid was working in cooperation with Seventeen or Bust on the Sierpinski problem and working with the Prime Sierpinski Project on the 'prime' Sierpinski problem. Although both Seventeen or Bust and the Prime Sierpinski Project have ceased operations, PrimeGrid continues the search independently to solve both conjectures.

The following k's remain for each project:

• Prime Sierpinski Project
  

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)
  

Recently discovered primes:

258317*2^5450519+1 is prime! Found by Sloth@PSP on July 28th, 2008.
90527*2^9162167+1 is prime! Found by Bold_Seeker@PSP on June 19th, 2010.
10223*2^31172165+1 discovered as part of our Seventeen or Bust subproject, eliminating 10223 from both the Sierpinski Problem and the Prime Sierpinski Problem, by Szabolcs Péter (SyP). (official announcement)
168451*2^19375200+1 is prime! Found by Ben Maloney (paleseptember) on September 17th, 2017. (official announcement)
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My lucky number is 75898⁵²⁴²⁸⁸+1

I know (2^n)-1 is prime when n is a prime

I'm sorry if this is a very naive question.

A Serpinkski number is defined as odd k where k*2^n+1 is not prime for any postive, odd integer n and 2^n>k see message 9502.

I can see that one can prove that a number is not a Serpinski number by finding a prime but surely to prove a number is a Serpinski number requires one to prove a negative. One can perhaps demonstrate that 78557*2^googolplex+1 +1 is not prime but that does not show that even larger numbers are composite. Or does it?

Be gentle with me on the maths and indeed errors of logic.

Cheers!

T

Thanks thommy3. I think that just about covers it. Mind you I might well come back with some more darn fool questions once I've sat done and worked through it but it looks like it will be a mighty fine start.

Again my sincere thanks.

T

Hello

The challenge I have finished 4 of my 10 pc not working on PrimGrid but its strange I still got sometimes wu psp sending on that 4 computers with BOINC 6.4.7 and 6.6.20 I can not understand wat it happening ?

Edit: Question, is it 100% certain that there will be a prime found? and if so could there be more then 1?

No, it's not 100% certain.
And there could be more than 1 (infinitely many) primes for each k. But only the first is of interest here.

Above when it says "some n must be found that makes k*2^n+1 prime." Doesn't that indicate that there is a 100% chance of a prime within the search numbers: ?

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.