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

Source: Romania TST 2016 Day 4 Problem 3

June 2, 2016
number theory

Problem Statement

Given a prime pp, prove that the sum ∑k=1⌊qp⌋kp−1\sum_{k=1}^{\lfloor \frac{q}{p} \rfloor}{k^{p-1}} is not divisible by qq for all but finitely many primes qq.
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