This looks like an interesting result! Wolfram alpha would be able to factor what you're looking for.
I was just wondering something
(p-1)! = -1 mod p
I was investigating the number (p-1)! + 1, which is divisible by p. I was wondering if n = ((p-1)! + 1)/p, then does the prime factorization of n consists of primes all of which are to the power of 1 (except for the case of p = 2, since n = 1)?
I couldn't really test much because of the factorial, it makes the numbers huge ... I tried till p = 37 ... after that I get problems with factoring the huge number generated by the factorial ...
Could someone test out a few more primes please ... or if someone is familiar with this idea, could you point me to more information about this ... thanks!
Thanks!! That's a neat site
Wolfram worked up to p=59 (except for p=47, not sure why, might be an internet problem on my part), after that, it doesn't work (at least up to p=101) ... but at least it's giving me hope. Everything I've tested so far seems to work
I was thinking that it might be a potential way to generate candidates for the largest prime , but then again, I'm not familiar with how it's usually done, hehehe.
Any other resources I could tap?
Good page on wikipedia which you might like to start at
Wilson's theorem - Wikipedia, the free encyclopedia
Don't think so either, the numbers don't seem deemed to be prime with high probability at all ... however their prime factorization is quite .. ah .. unusual. This has to be explored.I was thinking that it might be a potential way to generate candidates for the largest prime
Here's a mathematical headstart anyway :
Theorem
is not squarefree for prime .
Proof
Let be the greatest prime number that satisfies . Then for , thus . Since no exists for , we need .
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Maybe someone can put a condition on "if is not squarefree then is squarefree", which could potentially prove the conjecture if the condition is cleverly chosen ...
Wilson prime - Wikipedia, the free encyclopedia
Not exactly what you're after, but close.