Here is a simple theorem related to twin primes.

I wonder how many times it's been replicated.

If the positive integer d cannot be equal to abs( [ (3* m + 1) * n

+ ( 3* n + 1) * m ] )

for integers m and n, both nonzero,

then

both

6 * d -1 and 6 * d + 1 are prime.

Note that m and n are permitted to take on both positive and negative

values, but not zero.

Illustration:

m n (3*m+1)*n (3*n+1)*m sum

1 1 4 4 8

-1 -1 2 2 4

1 -1 -4 -2 -6

It's not possible for 1, 2, 3, or 5 to be of this form,

so both 6 * 1 - 1 and 6 * 1 + 1 are prime.

both 6 * 2 -1 and 6 * 2 + 1 are prime

both 6 * 3 -1 and 6 * 3 + 1 are prime

both 6 * 5 -1 and 6 * 5 + 1 are prime.

Note: It's necessary to exclude 0 from the permitted values of both m

and n because

If m = 0, and n is not zero, then

(3 * m + 1) * n + (3 * n + 1) * m = 1 * n + (3 * n + 1) * 0 = n, which

represents all integers.

Kermit Rose <

kermit@... >