Generalized taxicab numbers: pairs of squares While doodling with stuff on Mathematica, I noticed that, empirically, every generalized taxicab number to be representable as multiple sums of two (non-zero) squares is divisible by five:
$\forall n > 1 : 5 \mid \mathrm{Taxicab}(2, 2, n)$
($Taxicab$ defined in Generalized taxicab number.)
This observation is purely empirical, but seems to hold at least to taxicab numbers of value 1010. Some of the smallest sums of two squares that can be represented in $n$ distinct ways are:
$5^2 + 5^2 = 7^2 + 1^2 = 50$
$15^2 + 10^2 = 17^2 + 6^2 = 18^2 + 1^2 = 325$
$24^2 + 23^2 = 31^2 + 12^2 = 32^2 + 9^2 = 33^2 + 4^2 = 1105$
... and this goes on. For instance, $8800358125$ is the smallest such an integer that can be expressed $50$ different ways, and $6076533125$ can be expressed in $80$ ways.
I can't help thinking that this is a well known result. Is there a proof for this conjecture?
EDIT: I note that known $\mathrm{Taxicab}(3, 2, n)$ numbers (except the trivial case of $n = 1$) are divisible by $7$.

Edit: Thanks to A016032 and Ross Millikan's comment, the reason why appears to have been solved here by Joseph Culbertson. It follows Nate's answer and has to do with primes in the form $4k + 1$. 
Another link 
