(8k- 1) Let a prime number with the shape 8k- 1 be a finite r, that is, p 1, P2 ... let s = 8 (P 1 * P2 *...PR)? -1, and q is a factor of s, then (2/q) = (2 * s+2/q) = 1, the form of ∴ q is 8k 1, obviously the form of q is not 8k-1∴. But S=- 1(mod8) is contradictory, so the prime number in the form of 8k- 1 is infinite.

(8k+3) There are r primes with the shape of 8k+3: p 1, p2 ... vs. s = (2 * p1* p2 * ... pr)? +2, obviously there must be a prime factor Q that is an S in the form of 8k-3. (-2/q)=(S-2/q)= 1, but (-2/q) = (-1/q) * (2/q) =1* (1),

8k-3 Same as above