Problem

Source: VII Caucasus Mathematical Olympiad

Tags: algebra, number theory, geometric sequence



Positive integers $a$, $b$, $c$ are given. It is known that $\frac{c}{b}=\frac{b}{a}$, and the number $b^2-a-c+1$ is a prime. Prove that $a$ and $c$ are double of a squares of positive integers.