How can I found the smallest positive integer that can be represented as a sum of two positive square integers at least 64 ways? Here for example $5=1^2+2^2=2^2+1^2$ are two different representations of a sum. I found in http://mathworld.wolfram.com/SumofSquaresFunction.html that there is a function $r_2(n)$ but I think it is not suitable as it ignores order and signs. Is $5^3\cdot 13\cdot 17\cdot 29\cdot 37\cdot 41=1215306625$ the smallest integer satisfying the condition?
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$r_2(n)$ is suitable. There's a pretty simple relation between $r_2(n)$ and the number of ways to write $n$ as the sum of two squares of non-negative integers with the first not larger than the second. – Daniel Fischer Oct 26 '14 at 21:23
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Yes it is. Higher powers of $5$ (or any other of the prime factors) to drop large prime factors would make the number bigger, as would appending the next suitable prime $53$ and lowering the exponent of $5$. – Hagen von Eitzen Oct 26 '14 at 21:33
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The Brahmagupta–Fibonacci identity should get you started. A product of two sums of two squares is a sum of two squares in two different ways. Iterate the process and you can get more than two different ways.
For example, $(1^2+2^2)(2^2+3^2) = 4^2+7^2= 1^2+8^2$.
$$ (4^2+7^2)(1^2+3^2) = 25^2 + 5^2= 19^2 + 17^2 = 650 $$ $$ (1^2+8^2)(1^2+3^2) = 25^2 + 5^2 = 11^2 + 23^2 = 650 $$
One could say this gives you four ways to express $650$, but two of them are the same. The fact that some are the same as each other is a complication that will need to be dealt with.
