International Mathematics Competition
for University Students
2023

IMC2023: Day 1, Problem 4

Problem 4. Let \(\displaystyle p\) be a prime number and let \(\displaystyle k\) be a positive integer. Suppose that the numbers \(\displaystyle a_i=i^k+i\) for \(\displaystyle i=0,1,\ldots,p-1\) form a complete residue system modulo \(\displaystyle p\). What is the set of possible remainders of \(\displaystyle a_2\) upon division by \(\displaystyle p\)?

Tigran Hakobyan, Yerevan State University, Armenia

Hint: Consider \(\displaystyle \displaystyle\prod_{i=0}^{p-1}(i^k+i)\).

  Solution