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About the Prime Sierpinski Problem
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Michael GoetzVolunteer moderator Project administrator
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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:
Sierpinski problem 'prime' Sierpinski problem
21181 22699*
22699 67607*
24737 79309
55459 79817
67607 152267
156511
222113
225931
237019
* being tested as part of our Seventeen or Bust project
Fortunately, the two projects (and later PrimeGrid's Extended Sierpinski Project) combined their sieving efforts into a single file. Therefore, PrimeGrid's PSP sieve supports all three projects.
Additional Information
For more information about PSP, please see:
For more information about Sierpinski, Sierpinski number, and the Sierpinsk problem, please see these resources:
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 75898^{524288}+1
 


This is flippin' neat, this number theory stuff. It allows me to participate in advanced mathematics without knowing anything about it! :D
Now my question, directed at this thread:
Is there a substantive, actual, real use for Prime Sierpinski numbers, or is this an exercise in computing power? I know nothing of this advanced math, nothing at all; I know what primes are, and I know (2^n)1 is prime when n is a prime #. Beyond that, I know of no actual uses for prime numbers, although I hear they are used in cryptography in some form or fashion.
Basically, what will be advanced by learning what the smallest PS # is? Is it cryptography, probability, physics, something else, none of the above? I am curious... again, I have no math knowledge above Algebra II, so I'm learning here... :) John...
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Every battle is won before it is ever fought...
 


I know (2^n)1 is prime when n is a prime
Umm...
For n = 23, (2^23)1 = 8388607 which has a factor 47 so isn't prime.
If (2^n)1 is prime when n is a prime then the project would be redundant...
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35 x 2^3587843+1 is 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  


One can proof there is a pattern of factors repeating every 36 values on n. So all possible n are covered.
See http://www.teamprimerib.com/sob/78557.php for example.  


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 ?
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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 ?
Just a guess, but maybe you have different versions of boinc on your different computers??
Edit: Question, is it 100% certain that there will be a prime found? and if so could there be more then 1?
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John M. Johnson "Novex"  


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.  


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: ?
Sierpinski problem 'prime' Sierpinski problem 10223 10223* 21181 22699* 22699 67607* 24737 79309 55459 79817 67607 90527 152267 156511 168451 222113 225931 237019 ==?
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John M. Johnson "Novex"  

JohnHonorary cruncher
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Joined: 21 Feb 06 Posts: 2875 ID: 2449 Credit: 2,681,934 RAC: 0

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: ?
You left off the beginning of the quote which explains it all. In order to prove the conjecture, a prime must be found. This does not indicate that there's a 100% chance of finding a 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.
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