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a ^ 2 + b ^ 2 + 3, 2b ^ 2 + c ^ 2 + 3, 2c ^ 2 + a ^ 2 + 3 not perfect squares

Source: Mathematics Regional Olympiad of Mexico Center Zone 2019 P1

November 14, 2021
number theoryPerfect SquaresPerfect Square

Problem Statement

Let aa, bb, and cc be integers greater than zero. Show that the numbers 2a2+b2+3  ,2b2+c2+3  ,2c2+a2+32a ^ 2 + b ^ 2 + 3 \,\,, 2b ^ 2 + c ^ 2 + 3\,\,, 2c ^ 2 + a ^ 2 + 3 cannot be all perfect squares.
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