how to find taxicab numbers but for squares?

Question

Natural numbers that can be written as the sum of squares in two or more ways. The first ten numbers are 50, 65, 85, 125, 130, 145, 170, 185, 200, 205.

$ n = a^2 + b^2 = c^2 + d^2\\ a^2 − c^2 = d^2 − b^2\\ (a + c)(a − c) = (d + b)(d − b).$

Answer 1

Let $\Omega^*(n)$ denotes the number of primes $\equiv 1\pmod{4}$ that divide $n$, accounted with multiplicity. So we have, for instance: $\Omega^*(50)=2$ since $50=2\cdot 5^2$ and $\Omega^*(221)=2$ since $221=13\cdot 17$. Since every prime $\equiv 1\pmod{4}$ is a sum of two squares, and the set of integers that are sum of two squares are a semigroup, due to the Lagrange identity: $$ (a^2+b^2)(c^2+d^2)=(ac-bd)^2 + (ad+bc)^2,$$ every $n$ for which $\Omega^*(n)\geq 2$ is a "square taxicab number". The converse implication is also true, since the number of representations of an integer $m$ as a sum of two squares just depends on the number of divisors of $m$ that are $\equiv 1\pmod{4}$. An outstanding reference for this subject (and much more) is the David A.Cox book "Primes of the form $x^2+n y^2$".

Answer 2

The OEIS entry here references this writeup by D.J.C. Mackay and S. Mahjan entitled "Numbers that are Sums of Squares in Several Ways".