- Feb 5, 2009I was unable to find any web references other than Dirichlet's theorem.

Is the following statement obviously deducible from it or from some other

theorem or conjecture, or hard to believe, is it something new?

In any arithmetical progression of positive integers AP=a+n*b, where a

and b are odd positive integers that have no common divisor except 1,

there is a limitless number of pairs of successive prime terms P and Q,

such that R=P+Q-a and S=P+Q-b are consecutive primes.

The statement implies the existence of a limitless number of pairs of

consecutive primes whose difference equals any even integer. Would a

proof of it constitute a proof of the twin prime conjecture?

Here are some examples:

Take a=9 and b=25. The first occurrence of a pair of successive prime

terms P and Q is (36559, 36709), with R=73259 and S=73243. R and S are

consecutive primes with difference equal to 16. The second occurrence of

P and Q is (56359, 56509) with R=112859 and S=112843. It looks like one

can find as many occurrences as one's patience permits. My calculations

suggest that this is true for any (a, b) combination.

Take the reverse of the preceding example with a=25 and b=9. The first

occurrence of a pair of successive prime terms P and Q is (18133, 18169)

with R=36277 and S=36293. R and S are consecutive primes.

A large example. Take a=13 and b=171. The first occurrence of a pair of

successive prime terms P and Q is (196443103, 196443787) with R=392886877

and S=392886719. R and S are consecutive primes with gap=158.

Take the reverse of the above with a=171 and b=13. The first occurrence

of a pair of successive prime terms P and Q is (1789022887, 1789023043)

with R=3578045759 and S=3578045917. R and S are consecutive primes with

gap=158.

A very interesting example. Take a=7 and b=51. The first occurrence of a

pair of successive prime terms P and Q is (69877, 70183) with R=140053

and S=140009. R and S are consecutive primes with gap=44, and are the

same primes as the R and S primes in the next example where a and b are

reversed.

Take the reverse of the preceding example with a=51 and b=7. The first

occurrence of a pair of successive prime terms P and Q is (70009, 70051)

with R=140009 and S=140053. R and S are consecutive primes with gap=44.

Not every pair of consecutive primes can be the R and S consecutive

primes of some AP. I thought it would be interesting to see which primes

of Dr. Niceley's list of first occurrence gaps were R and S primes. Here

are the few I managed to get:

The first occurrence pair (89, 97) with gap 8 is the same as the R and S

prime pair resulting from the first occurrence of successive prime terms

P=47 and Q=53 in the AP with a=11 and b=3.

For first occurrence pair (139, 149) with gap 10, P=73, Q=79, a=13 and

b=3.

For first occurrence pair (5591, 5623) with gap 32, P=2659, Q=2971, a=7

and b=39.

For first occurrence pair (1669, 1693) with gap 24, P=839, Q=859, a=29

and b=5.

I would appreciate any comments. Thanks folks.

Bill Sindelar

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