My favorite fact in elementary number theory is that odd primes can be written as a sum of two squares if and only if they are congruent to 1 modulo 4. This has been known since 1634 and proven (by Euler) in 1747. However it is a very difficult fact of number theory that asymptotically half of all primes are congruent to 1 mod 4 (the other half being 3 mod 4); this is a consequence of Chebotarev’s density theorem [1], proven in 1922. Combining we find that asymptotically half of all primes are a sum of two squares.

What about numbers in general? It turns out that the asymptotic density of numbers which are a sum of two squares is 0. This follows from a theorem of Landau and Ramanujan in 1906 [2]. The proof appears not to be available online but there are some simple arguments that suggest the asymptotic density is 0.

So why the difference? Why are numbers of the form a^2 + b^2 more likely to be prime?

[1]  Wikipedia (and see also Wikipedia for fun)

[2]  Wikipedia

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