MathNet 2024-part-04 — 168 problems (EN–VI) Question 167

Let $k \ge 2$ be a given positive integer. Determine the number of pairs $(m, n)$ of positive integers, with $m, n \lt 2^k$, such that the equation $z^{2^k} - z^m - z^n = 1$ has at least one solution with modulus $1$.

Bogdan Blaga