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).

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