There are three kinds of prime numbers. The first is a solitary outlier: 2, the only even prime. After that, half the primes leave a remainder of 1 when divided by 4. The other half leave a remainder of 3. (5 and 13 fall in the first camp, 7 and 11 in the second.) There is no obvious reason that remainder-1 primes and remainder-3 primes should behave in fundamentally different ways. But they do.