One, For all pairs of odd positive primes A<B with the same difference D and A>3, the number N of even integers between A and B that are divisible by a twin prime middle number remains the same, and equals D/3, if D mod 3 equals 0, or equals (D-1)/3, if D mod 3 equals 1, or equals (D+1)/3, if D mod 3 equals 2.

I found statement one to be true for many (A, B) prime pairs.

I also found the following generalized statement two to be true for many non-prime odd integers.

Two, For all pairs of odd positive integers A<B, that satisfy these 4 requirements:

a), Their greatest common divisor equals 1.

b), They are not divisible by 3.

c), They are not prime

d), They have the same difference D.

the number N of even integers between A and B that are divisible by a twin prime middle number remains the same, and equals D/3, if D mod 3 equals 0, or equals (D-1)/3, if D mod 3 equals 1, or equals (D+1)/3, if D mod 3 equals 2.

Statement one example: Pick D=22. The first occurrence of a pair of primes with a difference D=22 is A=7 and B=29. The number N of even integers between A and B that are divisible by a twin prime middle number is N=7 by actual count. Since (D mod 3) equals 1, (D-1)/3=7 which agrees with the actual count N. Take the consecutive primes A=1129 and B=1151. D=22. Again N equals 7, and remains constant for all pairs of odd primes with the same difference D=22.

Statement two example: Again pick D=22. The first occurrence of a pair of odd integers with a difference of 22 that satisfies the 4 requirements is A=133 and B=155. Again N==7. prompting the following statement three.

For convenience, let’s call any pair of distinct odd primes a “P”, and any pair of distinct odd positive integers that satisfies the 4 requirements a “Z”.

Three, if a P and a Z have the same difference D, then the number Np of even integers in P that are divisible by a twin prime middle number always equals the number Nz of even integers in Z that are divisible by a twin prime middle number.

Is all of the above obvious and well known? Anyone care try these exercises and check me on this? Thanks folks.

Bill Sindelar

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