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Solutions to |px^2-qy^2|=1

Source: Kürschák József Competition 2022/2

October 7, 2022

number theoryprime numberssquare

Problem Statement

Let

p

and

q

be prime numbers of the form

4k+3

. Suppose that there exist integers

x

and

y

such that

x^2-pqy^2=1

. Prove that there exist positive integers

a

and

b

such that

|pa^2-qb^2|=1

.

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