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a- b, 2a + 2b + 1, 3a + 3b + 1 perfect sqaures if 2a^2 + a = 3b^2 + b

Source: Chile Finals 2010 L2 p1

October 5, 2022
number theoryPerfect Square

Problem Statement

The integers a,ba, b satisfy the following identity 2a2+a=3b2+b.2a^2 + a = 3b^2 + b. Prove that aāˆ’ba- b, 2a+2b+12a + 2b + 1, and 3a+3b+13a + 3b + 1 are perfect squares.