Four different positive integers a, b, c, and d are such that $a^2 + b^2 = c^2 + d^2$ Four different positive integers $a, b, c$, and $d$ are such that $a^2 + b^2 = c^2 + d^2$
What is the smallest possible value of $abcd$?
I just need a few hints, nothing else. How should I begin?  Number theory?

The number of representations of a positive integer as a sum of two squares depends on the number of prime divisors of the form $4k+1$ (see Cox, Primes of the form $a^2+k b^2$). If we take the first two primes of such a form, $5$ and $13$, we have that $5\cdot 13$ can be represented as:
$$ 65 = 1^2+8^2 = 4^2+7^2 $$
so we have a solution with $abcd = 224$. You can complete the proof by exhaustive search (it is quite easy to check that the first $64$ positive integers do not have a double representation in terms of positive integers, and we just have to check till $n=225$ or so to find the minimum $abcd$), or proving that the number of representations of $n$ as a sum of two squares is given by the number of divisors of $n$ of the form $4k+1$ minus the number of divisors of the form $4k+3$.
