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Double of a squares in geometric progression

Source: VII Caucasus Mathematical Olympiad

March 13, 2022

algebranumber theorygeometric sequence

Problem Statement

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.

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