International Mathematics Competition
for University Students
2020

IMC2020: Day 2, Problem 6

Problem 6. Find all prime numbers \(\displaystyle p\) for which there exists a unique \(\displaystyle a\in \{1,2,\ldots,p\}\) such that \(\displaystyle a^3-3a+1\) is divisible by \(\displaystyle p\).

Géza Kós, Loránd Eötvös University, Budapest

Hint: Compute the roots of \(\displaystyle f(x)=x^3-3x+1\) over the complex field and find a connection between the roots.

Alternatively, compute the discriminant of \(\displaystyle f(x)\).

  Solution