While reading up on Pythagorean quadruples and Legendre's three square theorem, I encountered the following problem:
Since there are an infinite number of Pythagorean quadruples, it is true that the equation $a^2 + b^2 + c^2 = d^2$ has an infinite number of positive integer solutions. For example, $3^2 + 4^2 + 12^2 = 13^2$. In the same sense, we can show that $5^2$ can be written as the sum of two squares $3^2 + 4^2$ and the difference of two other squares $13^2 -12^2$. Using the results above, is it possible to show that any perfect square $k^2$ can be simultaneously written as the sum of two squares $a^2 + b^2$ and the difference of two other squares $d^2 - c^2?$
For a different version of the question, click here: Range of values of $k^2$ equal to the sum of two squares and the difference of two other squares.