MathDB

S_k=1^k+2^k+...+(p-1)^k

Source: Moldova TST 2012

March 9, 2023

number theory

Problem Statement

Let

p\geq5

be a prime and

S_k=1^k+2^k+...+(p-1)^k,\forall k\in\mathbb{N}.

Prove that there is an infinity of numbers

n\in\mathbb{N}

such that

p^3

divides

S_n

and

p

divides

S_{n-1}

and

S_{n-2}.