Hi All,

I just bought a new crunching rig with a decent graphics card, and decided to put it to work on the AP27 project. Looking at the first few results I found yesterday and today, and comparing with an ancient AP21 I found here at Primegrid 9 years ago (until I switched my new PC on yesterday this AP21 was the only AP I had ever found) I noticed that one common feature they all share is that the prime sequences are constructed by multiplying a given quantity by 23#. For example:

11988702023996197+4465007*23#*n for n=0..20

This made me curious as to whether there is any significance in the choice of 23# here (since it seems not to have changed in 9 years) - are any other sequences, constructed of say 29# or 17#, being tested as well? For numbers of the form a+b*c#*n for some range of n, it occurs to me we could get similar densities of primes and similar AP's as we are testing now, but using a different value than c=23.

Thanks
Darryl
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In an arithmetic progression of form An + B, a prime p which does not divide the common difference A, will divide every p-th term of the progression.

For example, in any progression of form 60n + B, no matter what B is, every seventh term is divisible by the prime 7. That is because 7 does not divide 60. So if you want an AP that is greater than 7 in length (or just greater than 6 if no term of the AP equals 7 itself), you want a coefficient A which is divisible by 7.

If you want an AP of length 23, 24, 25, 26, 27, or 28, then the A in the formula An + B must be divisible by all the primes 2, 3, 5, 7, 11, 13, 17, 19, and 23.

This explains the 23# you see.

When, in the future, someone searches for an AP29 (and one not starting from the term 29, so B≠29), they would need to substitute 23# with 29#.

/JeppeSN

Thanks for both of these answers. It makes much more sense now!

Cheers,
Darryl
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