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December 30th, 2007, 09:35 PM   #1
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Wieferich primes - searching on www.elmath.org

You can participate in the project Wieferich@home aimed at searching for Wieferich primes.

Download free application on www.elmath.org.

Wieferich primes satisfy
2^(pāˆ’1) ā‰” 1 (mod p^2)
and only two Wieferich primes are known up to now: 1093 and 3511.
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January 4th, 2008, 12:06 AM   #2
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Dude there aren't any more - I checked.

(j/k)
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January 14th, 2008, 10:25 PM   #3
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Did You check it for all primes? Brilliant!
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January 15th, 2008, 04:57 AM   #4
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So, tell us about the project -- how do you find Wieferich primes, or more to the point, how do you quickly discard composite candidates?


brunojoyal, you know that doesn't help... if the problem is outside of NP (well, really outside the Arthur-Merlin protocol) then *your* infinite computational resources don't help *us* solve the problem, since we can't verify that your calculations are correct.
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January 15th, 2008, 05:32 AM   #5
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Actually we prepare new version with a quick implementation of the sieve of Eratosthenes and our original modular exponentiation. A project is live and always open to good advice.
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February 5th, 2008, 04:16 AM   #6
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The notably increasing speed of searching and additional innovations are in Version 2. More than 200 users ā€“ participate!
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February 5th, 2008, 09:21 AM   #7
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Quote:
Originally Posted by miroslavkures
Actually we prepare new version with a quick implementation of the sieve of Eratosthenes and our original modular exponentiation. A project is live and always open to good advice.
Very nice. I wasn't able to find any documentation on the site about the inner workings of the software, though.
  • Algorithms: are you simply testing each prime mod p^2, or is there some better method? How is exponentiation done -- binary, ladder, something else?[/*:m:q6vu0wm1]
  • How do you work with large numbers -- an existing library or custom-rolled? Or do you avoid working with numbers with more than 64 bits?[/*:m:q6vu0wm1]
  • Is the source available somewhere, or even just key parts? This would help explain some algorithms even without documentation. Is the source portable? Does it have fast assembly routines for some processors?[/*:m:q6vu0wm1]
  • What ranges are you searching? Is there any overlap with other projects?[/*:m:q6vu0wm1]
  • Is there any double-checking done? Are checksums computed and reported, or only progress?[/*:m:q6vu0wm1]
  • Do you know of any interesting theoretical results relating to Wieferich primes? All I know about is that abc implies that there are infinitely many non-Wieferich primes... but has anyone shown something stronger, like positive upper density of non-Wieferich primes under the abc conjecture?[/*:m:q6vu0wm1]
  • Have their been any recent discoveries regarding Wieferich primes?[/*:m:q6vu0wm1]
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