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ax^2 + by^2 = 1 infinite rational solutions if it has one

Source: 1978 Hungary - Kürschák Competition p1

October 15, 2022

algebranumber theoryrational

Problem Statement

a

and

b

are rationals. Show that if

ax^2 + by^2 = 1

has a rational solution (in

x

and

y

), then it must have infinitely many.