MathNet 2024-part-04 — 168 problems (EN–VI) Question 157

Let $(A, +, \cdot)$ be a finite ring and $a, b \in A$ with $ab + ba = 0$. If there is an $n \in \mathbb{N}^*$, $n \not\equiv 2 \pmod 4$, such that $a^n - b^n$ is invertible, prove that $a - b$ and $a + b$ are invertible.

Mihai Opincariu