Number theory on square numbers

Question

Can a square number be split as the sum of two other squares in two different ways? The Pythagorean numbers help us to identify numbers which can be split as sum of two squares. Is this unique?

Answer 1

HINT

$3,4,5$ is a PPT, meaning that any multiple (like $6,8,10$) is a PT

$5,12,13$ is a PPT as well ...

Answer 2

For example \begin{eqnarray*} 65^2=63^2+16^2=33^2+56^2. \end{eqnarray*}

Answer 3

By the Brahmagupta–Fibonacci identity, which states that $$(a^2 + b^2)(c^2 + d^2) = (ac-bd)^2 + (ad+bc)^2 = (ac+bd)^2 + (ad-bc)^2$$ you can get a number which is a sum of two squares two different ways by multiplying two sums of two squares.

So from two Pythagorean triples $a^2 + b^2 = a_1^2$ and $c^2 + d^2 = a_2^2$ , the left hand side simplifies to $(a_1a_2)^2$, you can get a number that can be written as a sum of two squares.