If k is a positive integer such that there does not exist a nonzero m,

different than k, such that

(6 m + 1) divides (k-m), then (6k+1) is prime.

2:

If k is a positive integer such that there does not exist a nonzero m,

different than (-k), such that

(6m + 1) divides (k+m), then (6k-1) is prime.

Kermit