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Old 2007-06-25, 19:22   #1
MatWur-S530113
 
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Apr 2007
Spessart/Germany

2428 Posts
Red face Is this to prove/already known?

Hello,

I got this conjecture last night, but my math is not good enough to prove it...

if and only if n^{n-1} is divisible by ((n-1) / 2)^2 then there is a non-negative Integer k with n=2^k+2 (= M(k) + 3)

another conjecture:
if n>1, n odd then n^{n-1}-1 is divisible by ((n-1)/2)^2

are these conjectures already known/proven? And if not, any idea how to prove them?
I got them while looking in a self-created file with the (start of) factorisations of numbers of the form: n=p^{p-1}-1, p prime (I made TF in this file up to 30 bit, if wanted I'll post it).

Thanks in advance.

mfg

Matthias
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