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Asking my mathsy people ...

An odd prime is the sum of two squares iff it's 1 (mod 4).

That means that 29 (say) factors in C.

There exists a square root of -1 mod p iff it's 1 (mod 4) (Again, limited to odd primes).

That means that 29 (say) factors non-trivially mod 41 (say).

These are all clearly related, and there are connections to be had, some obvious, others likely less so.

Does anyone know of a good article that gives a feel for what's happening?

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@ColinTheMathmo Both ℂ and ℤ/41ℤ are fields, so saying that things "factor nontrivially" in either of those systems... is sort of true, depending on what you mean by nontrivial, but not really the point. Everything factors lots of ways in a field, because everything's a unit. For every nonzero x, there is a y such that 29=x*y (mod 41), and that's nothing special about 29. It isn't really taking advantage of the fact that 9² = -1 (mod 41).

In the first case, the more relevant fact is nthat 29 factors in ℤ[i], the Gaussian integers. (If that's what you meant, then apologies for overexplaining; I misunderstood your notation.) Viz., (5+2i)(5-2i). This is a ring with interesting factorizations because not everything is a unit. I can't think of any ring that uses the "square root of -1" in ℤ/41ℤ in the same way.

@ColinTheMathmo What I don't know, or can't bring to mind, is whether "-1 is a quadratic residue mod p" and "p is a sum of two squares" are related in a direct and important way or if it's really just a coincidince that "p is congruent to 1 mod 4" characterizes both. The usual proofs I know for those two things don't seem to have much to do with each other but maybe I'll learn something here.

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