[seqfan] On equivalence of congruences

[Tomasz Ordowski]

Dear SeqFan,

Conjecture:

For a prime p;
1^(p-1)+2^(p-1)+...+n^(p-1) == n (mod p)
if and only if
binomial(n+p,p) == 1 (mod p).

I am asking for proof or counterexample.

Cf. https://oeis.org/A133907

? a(n) is the smallest prime p such that
1^(p-1)+2^(p-1)+...+n^(p-1) == n (mod p).

Best regards,

Thomas
_________
For n = p-1, binomial(2p-1,p) == 1 (mod p^3) with p>3.

https://en.wikipedia.org/wiki/Wolstenholme%27s_theorem

https://en.wikipedia.org/wiki/Agoh%E2%80%93Giuga_conjecture




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