There is a huge (maybe infinite) number of twin primes, like (11,13), but only one triplet with d=2, i.e. (3,5,7).
However, one can get longer arithmetic series with other values of d.
Please list all the increasing arithmetic series such that:
a. They consist of at least 5 prime members.
b. Each number is under 100.
Example: 5,11,17,23,29. (d=6)
A computation shows that the smallest possible common difference for a set of <IMG class=inlineformula height=14 alt=n src="http://mathworld.wolfram.com/images/equations/PrimeArithmeticProgression/Inline42.gif" width=7 border=0> or more primes in arithmetic progression for <IMG class=inlineformula height=14 alt=n=1 src="http://mathworld.wolfram.com/images/equations/PrimeArithmeticProgression/Inline43.gif" width=31 border=0>, 2, 3, ... is 0, 1, 2, 6, 6, 30, 150, 210, 210, 210, 2310, 2310, 30030, 30030, 30030, 30030, 510510, ... (Sloane's A033188, Ribenboim 1989, Dubner and Nelson 1997). The values up to <IMG class=inlineformula height=14 alt=n=18 src="http://mathworld.wolfram.com/images/equations/PrimeArithmeticProgression/Inline44.gif" width=38 border=0> are rigorous, while the remainder are lower bounds which assume the validity of the prime patterns conjecture and are simply given by <IMG class=inlineformula height=14 alt=n# src="http://mathworld.wolfram.com/images/equations/PrimeArithmeticProgression/Inline45.gif" width=20 border=0>. The smallest first terms of arithmetic progressions of <IMG class=inlineformula height=14 alt=n src="http://mathworld.wolfram.com/images/equations/PrimeArithmeticProgression/Inline46.gif" width=7 border=0> primes with minimal differences are 2, 2, 3, 5, 5, 7, 7, 199, 199, 199, 60858179, 147692845283, 14933623, 834172298383, ... (Sloane's A033189; Wilson).
More about arithmetic progressions with primew
