The Problems of Ukrainian Authors

Question

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.

Accepted Answer

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 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.