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Old 2015-05-19, 12:17   #1
arbooker's Avatar
"Andrew Booker"
Mar 2013

5516 Posts
Default OEIS A071580: Smallest prime of the form k*a(n-1)*a(n-2)*...*a(1)+1

I have been working on extending OEIS sequence A071580, whose nth term is the smallest prime congruent to 1 mod the product of all smaller terms. The sequence enjoys several nice properties in common with the sequence of Mersenne primes:

1) There is a fast algorithm to prove primality: since p-1 has a prime factor of about square root size, the simplest variant of Pocklington's criterion runs about as quickly as a Fermat PRP test.

2) It has rapid growth (doubly exponential), so we get to big primes quickly.

3) It has an intrinsic definition, as opposed to (say) Proth primes, which are a bit quicker to test, but for which the choice of numbers is completely ad hoc.

I have computed the first 23 terms, the last of which is a prime with over a million digits. I used gwnum to find the terms and double-checked the first 22 of them with gmp.

It took a lot of effort to find the 23rd term (it was roughly a factor of 3 larger than expected), and I'm thinking of stopping at this point, but if any of you guys with substantial resources would be interested in helping to find the 24th term or double-check the 23rd, let me know. There is a reasonable chance of finding the 24th term (which will be a prime with over 2 million digits) within a year.
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