[seqfan] Re: more

[Ami Eldar]

 ((M+1)^M - 1) / ((2^M-1)M^2) with M=2^13-1 is composite.
The next candidate, ((M+1)^M - 1) / ((2^M-1)M^2) with M=2^17 is a huge
number with 631291 digits.

Best regards,
Amiram


On Sat, Sep 18, 2021 at 12:21 PM Tomasz Ordowski <tomaszordowski at gmail.com>
wrote:

> Hello!
>
> Note that m^2 | (m+1)^m - 1 for every m > 0.
> I noticed that if m = 2^n-1, then 2^m-1 | (m+1)^m - 1.
> So if m = 2^n-1 and (m, n) = 1, then (2^m-1)m^2 | (m+1)^m - 1.
> Hence, if m = 2^n-1 is squarefree, then (2^m-1)m^2 | (m+1)^m - 1.
> In particular, this divisibility holds for Mersenne numbers. Let's define:
>    Numbers M = 2^p-1 such that ((M+1)^M - 1) / ((2^M-1)M^2) is prime.
> Such M must be a Mersenne prime. I found 2^3-1 = 7 and 2^7-1 = 127.
> Maybe someone will find more (the "next" prime M = 2^127-1 is nice /;-).
>
>
> Regards!
>
> T. Ordowski
> _____________
> A060073 - OEIS <https://oeis.org/A060073> (a(n+1) = ((n+1)^n - 1)/n^2).
> A127837 - OEIS <https://oeis.org/A127837> (cannot be Mersenne primes > 3).
>
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