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A non-constant AP with square of terms also as terms of AP

Source: INMO 2016 Problem 6

January 17, 2016

arithmetic sequencenumber theoryrelatively primenumber theory proposed

Problem Statement

Consider a nonconstant arithmetic progression

a_1, a_2,\cdots, a_n,\cdots

. Suppose there exist relatively prime positive integers

p>1

and

q>1

such that

a_1^2, a_{p+1}^2

and

a_{q+1}^2

are also the terms of the same arithmetic progression. Prove that the terms of the arithmetic progression are all integers.