## Thursday, October 06, 2016

### [sxbpxqpu] Splitting the primes by 3, 4, 6

All prime numbers except 2 are of the form 4k+1 or 4k+3.  (Actually, all odd numbers.)  This distinction is made famous in Fermat's Sum of Squares theorem and quadratic reciprocity.

All prime numbers except 3 are of the form 3k+1 or 3k+2.

All prime numbers except 2 and 3 are of the form 6k+1 or 6k+5.

Each of the residue classes contains roughly half the prime numbers.  (Nice article: "Prime Number Races" by Granville and Martin.)  I don't think there are other moduli which result in two classes of equal size.  (All prime numbers (all numbers) are of the form 2k+0 or 2k+1, but former contains only prime element, namely 2.)

The remainders for each of these can also be written +1 and -1.  Therefore, each prime can be annotated with 3 signs, moduli 3, 4, and 6.

The annotation for currently largest known Mersenne Prime M74207281 is +-+ (plus minus plus).  The annotation for the exponent is +++ (plus plus plus).