Solving $x^2 + y^2 = M$ with $M$ prime

Question

This question is related with the topic: Solving $x^2 + y^2 = M$ over integers

I have understood the formula used to determine $(x,y)$ such that $x^2+y^2=M$ with $M$ composite. But what if $M$ is a large prime such as $M=1000000241$ do I have to do only bruteforcing? Are there other methods?

EDIT:

I know that $M$ must be a prime of the form $4k+1$ but how to find $(x,y)$?

Thanks

For any such numbers we can get an answer immediately. Think about this. — Michael Rozenberg, Jan 12 '20 at 17:54
I changed the number to an actual prime (as suggested by OP’s original statement). — Kieren MacMillan, Jan 12 '20 at 17:55
This is Fermat theorem that primes can be written as sum of two squares if and only if $p=4k+1$. Your M satisfies this condition. — sirous, Jan 12 '20 at 18:17
https://math.stackexchange.com/questions/5877/efficiently-finding-two-squares-which-sum-to-a-prime is a prior question which discusses some methods in the answers. — JoshuaZ, Jan 12 '20 at 18:49
Does this answer your question? Efficiently finding two squares which sum to a prime — JoshuaZ, Jan 12 '20 at 18:49

score 1 · Answer 1 · answered Jan 12 '20 at 17:58

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The bad news is that, yes, brute force is [currently] your only real recourse for large primes.

The good news is that primes which are the sum of two squares have a unique representation in that form… so once you find a representation, you’re done.

answered Jan 12 '20 at 17:58

Kieren MacMillan

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And what for a smaller prime $M$? – Matteo Jan 12 '20 at 17:59
you can actually sieve out a lot. it's just you can't tell which part is which one must be nonzero mod 3, one must be odd, one must be 1 or 4 mod 5... etc. – Jan 12 '20 at 18:20

score 0 · Answer 2 · answered Jan 12 '20 at 18:09

$x^2 + y^2 = M$ represents a circle with $(0,0)$ as its center and $\sqrt{M}$ as its radius.

So once you plot this circle on a Cartesian system, then you should be able to find all values of $x$ and $y$ that satisfy your equation.

(I'm assuming you're allowed the use of a graphing calculator.)

Solving $x^2 + y^2 = M$ with $M$ prime

2 Answers2