Competition problem — Number theory

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Number theory Difficulty 4.6 AIME Prove it Ukraine

Prove, that the equation $a^3 + b^3 + c^3 = a^2 + b^2 + c^2$ has infinite number of solutions in the integer numbers, if the biggest common divisor of numbers $a, b, c$ is 1.

Solution

Lets choose such values: $b = 1 + x$ , $c = 1 - x$ and put it in desired equation:
$a^3 + 2(1 + 3x^2) = 2(1 + x^2) + a^2 \Leftrightarrow a^2 - a^3 = 4x^2.$
We put $1 - a = 4p^2$ , then
$a^2(1 - a) = (4p^2 - 1)^2 4p^2 = (2x)^2.$
We put in a number $x = p(4p^2 - 1)$ . Then three integer numbers $(a, b, c) = (1 - 4p^2, 1 + p(4p^2 - 1), 1 - p(4p^2 - 1))$ satisfy equation, i.e. there are infinite number of solutions.

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