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I was trying to find the numbers of possibilities of $x^2+y^2$ that are not perfect squares for $0\le x\le m$ and $0\le y\le n$ and $x+y>0$.

I think if we calculate the numbers of possibilities of $x^2+y^2=d^2$, it will do it. Any formulas or even an algorithm?

JohnD
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Maggi Iggam
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  • One should not expect an exact closed form expression. There is a pleasant formula for the number of integer solutions of $x^2+y^2=d^2$, in terms of the prime power factorization of $d$. – André Nicolas Jan 28 '13 at 16:40
  • @AndréNicolas : thanks , please tell that formula still though i just wanted a formula for any d :) – Maggi Iggam Jan 28 '13 at 16:44
  • @MaggiIggam Note that the formula does not take into consideration the bounds on $x$ and $y$, so it won't necessarily be exact for $d \ge \min{m,n}$. Nevertheless this can be turned into an algorithm with the potential to be more efficient than $O(mn)$ brute force (but much harder to implement). – Erick Wong Jan 28 '13 at 16:54

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We give a count of the number of representations of an integer as a sum of two squares, if there are no size restrictions. This does not answer the question, but is in response to a request from the OP.

Let $n$ be a positive integer, and let $$n=2^ap_1^{a_1}\cdots p_s^{a^s} q_1^{b_1} \cdots q_t^{b_t},$$ where the $p_i$ are distinct primes of the form $4k+1$, and the $q_j$ are distinct primes of the form $4k+3$.

If at least one of the $b_j$ is odd, then $n$ has no representations as a sum of the squares of two integers.

If all the $b_j$ are even, then $n$ has $N(n)$ representations as the sum of the squares of two integers (positive, negative, or $0$), where $$N(n)=4(a_1+1)(a_2+1)\cdots(a_s+1).$$

We may instead be interested in the number of essentially distinct representations, where for example the representations $2^2+3^2=13$, $3^2+2^2=13$, $(-2)^2+3^2=13$, and so on are considered to be "the same." Note that if $n=k^2$, the representation $n=k^2+0^2$ is included in the count.

If $8$ divides $N(n)$, then there are exactly $\dfrac{N(n)}{8}$ essentially distinct representations of $n$ as a sum of two squares.

If $8$ does not divide $N(n)$, then there are $\dfrac{N(n)+4}{8}$ essentially distinct representations of $n$ as a sum of two squares.

Now let $n=d^2$. If we know the prime power factorization of $d$, we can write down the prime power factorization of $d^2$, and then use the above formulas.

André Nicolas
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  • Nice elaborate answer, +1. Note that the condition "$8$ does not divide $N(n)$" means that all exponents in the factorisation of $n$ must be even, except possibly the exponent $a$ of $2$; therefore this happens if and only if either $n$ is a perfect square (in which case $n=k^2+0^2$ is one of the solutions) or $n$ is twice a perfect square (in which case $n=k^2+k^2$ is one of the solutions). In particular it is this first special case that applies concretely to the question. Note that the other special case was evoked in this question. – Marc van Leeuwen Jan 29 '13 at 11:29