Find all positive integers x > y > z such that x²+27y, y²+27z, z²+27x and x²+y²+z²+27 are all perfect squares.

With the aid of a computer program, I can confirm that 108, 75, 72 is a solution, and the only solution for x <= 1000. But I don't have a proof that there are no others.

Posted by Paul on 2019-01-23 20:15:42