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)

(In reply to computer solution by Charlie)

If we allow all numbers below 200 instead of below 100 we get the following:

 series                        d
 5  11  17  23  29             6
 5  17  29  41  53             12
 5  47  89  131  173           42
 5  53  101  149  197          48
 7  37  67  97  127            30
 7  37  67  97  127  157       30
 11  41  71  101  131          30
 37  67  97  127  157          30
 

Note that as there is a 6-member series, the two 5-member series that contain the same elements save the first or last also appear.

We can expand even further, and look for longer series. Looking for series of at least 7 members and going up to the 300th prime (1987), the following series are found:

 series                                                 d
 7  157  307  457  607  757  907                       150
 47  257  467  677  887  1097  1307                    210
 179  389  599  809  1019  1229  1439                  210
 199  409  619  829  1039  1249  1459  1669  1879      210

Here I've deleted the proper subsets of the 9-member series.

Posted by Charlie on 2011-11-17 12:53:04