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MatWur-S530113 · 2007-06-25, 19:22

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

2007-06-25, 19:22	#1
MatWur-S530113 Apr 2007 Spessart/Germany A2₁₆ Posts	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