All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
 perplexus dot info

 Twins, triplets and more (Posted on 2011-11-17)
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)

Comments: ( Back to comment list | You must be logged in to post comments.)
 expanding (spoiler included) | Comment 2 of 5 |
(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

 Search: Search body:
Forums (0)