N2

Part of 2016 Romanian Master of Mathematics Shortlist

Problems(1)

Prove that sum is not divisible by q for finitely many q

Source: Romania TST 2016 Day 4 Problem 3

p

, prove that the sum

\sum_{k=1}^{\lfloor \frac{q}{p} \rfloor}{k^{p-1}}

is not divisible by

q

for all but finitely many primes

q