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I am trying to find all of the answers to $r_2(n^2) = 420$, where $N < 10^{11}$. It is for finding lattice points on a circle with points $(0,0), (N,0), (0,N)$, and $(N,N)$. I am (pretty) sure that all of the following answers work:

359125, 469625, 612625, 718250, 781625, 866125, 933725, 939250, 1047625, 1119625, 1225250, 1288625, 1336625, 1366625

where $r_k(n)$ is the number of different squares $k$ (in this case, the sum of 2 different squares) that add up to $n$. (Source)

I have noticed that all of these are divisible by $5^2$, but I know that the answer to this problem ends in 309.

What kinds of numbers would get me 420 for this?

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    It did come from PE. I hope what I have asked isn't considered cheating.2012-03-06

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Hint: use the following well-known formula

$\qquad\quad n = 2^{c} p_1^{a_1}\:\cdots\:p_r^{a_r} q_1^{b_1}\:\cdots\:q_s^{b_s}\quad$ where primes $\ p_i \equiv 3,\ q_i \equiv 1\pmod{4}$

$\quad\ \Rightarrow\ {\rm SquaresR}[2,n] = 4\:(b_1+1)\:\cdots\:(b_s+1)\:$ if $a_i$ are all even, else $0$

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    This was in reply to the first version of your question, which did not have the link. Apply the formula and solve for the $b_i$.2012-02-12