Talk:Consecutive primes in arithmetic progression

10000024493

I don't get this:

In 1967, Jones, Lal and Blundon found five consecutive primes in arithmetic progression: ($10^{10}+24493+30k,\,k\,=\,0,1,2,3,4\,$).

Those numbers are $9843019+30k$ for $k=0,1,2,3,4$. Where does the $10^{10}$ come from? --Prime Mover 06:05, 12 October 2017 (UTC)

Let's see, REXX numeric digits 20; X=10**10 + 24493; S=X; do K=1 to 4; S=S || ',' || X+30*K; end K; say S yields 10000024493,10000024523,10000024553,10000024583,10000024613. The OEIS does not know it, it might be a NOGI full. The OEIS also does not know 10000024493, it's too big for A052243. I've added A052243 as XREF to A006560. –Frank Ellermann 04:44, 13 October 2017 (UTC)

The numbers are correct. $10^{10}+24493+30k,\,k\,=\,0,1,2,3,4\,$ was (probably) the first discovery of five consecutive primes in arithmetic progression. It's mentioned at https://primes.utm.edu/top20/page.php?id=13. Smaller primes were discovered later. $9843019+30k$ is the smallest case. The page doesn't claim it's the same case. Jens Kruse Andersen 12:47, 13 October 2017 (UTC)

Hi, I changed my mind later and submitted draft/A293791, 1967 and in a book 2009 and on the page here are not exactly "NOGI". Actually the page should be linked if the quintuplet survives its review and if the CPAP page reaches some "permitted OEIS link target" state. –Frank Ellermann 15:02, 16 October 2017 (UTC)

Update: A293791 made it, now trying to get a nonn,hard,more instead of only nonn in A126989, hopefully somebody throws in a nice IFF A126989(7)=210=7*5*3*2*1=4#=A002110(4) is verified. –Frank Ellermann 23:54, 21 October 2017 (UTC)

nonn,hard,more,changed: Shoot for nice, good hunting! –Frank Ellermann 04:53, 22 October 2017 (UTC)

a(7) > 10^21 (see link)

That's a case of Error: The file was not found, i.e., which link, there are lots of links on the page, and the "ten" paper referenced in A006550A006560^{see below} mumbles something about "[f]or k = 7 this product is about 10^59" on page 2. That paper also uses 5# for 5*3*2*1 instead of 3# for p(3)*p(2)*p(1)*p(0), but something with the primorials on the page here was anyway wrong, I used the notation on primorial matching <URL:http://mathworld.wolfram.com/Primorial.html>. Likewise "it is conjectured"  doesn't tell me who, where, and when, meanwhile I think that it was Guy decades ago, IOW, no nonsense. –Frank Ellermann 23:43, 21 October 2017 (UTC)

You mean A006560 and not A006550. I guess "The expected size is a(7) > 10^21 (see link)" refers to my own site http://primerecords.dk/cpap.htm#smallest which says: "Heuristics (estimates based on probability) indicate the minimal CPAP-7 may have 22 or 23 digits. The smallest known is 32 digits". I don't show or reference the calculations for "22 or 23". 32 is a proven upper bound so it certainly isn't near 10^59 which is an unrelated number. 5# does mean 5*3*2. 3# means 3*2 and cannot mean p(3)*p(2)*p(1). The notation p₃# (where the letter p is part of the notation and not a variable) means p(3)*p(2)*p(1). p₃ = 5, so p₃# = 5#. "It is conjectured that there are arithmetic progressions of n consecutive primes, for any n." This follows from the widely believed prime k-tuple conjecture [1] that all admissible prime tuples have infinitely many occurrences. That conjecture doesn't mention consecutive primes but for any admissible tuple you can construct another admissible tuple with more members such that the original tuple is a sub-tuple and must have consecutive primes because all other numbers in the interval are forced to have selected prime factors. --Jens Kruse Andersen 02:22, 22 October 2017 (UTC)

Fixed before looking here, I just quoted your statement, {{cite web}} is flexible. If you like the other n# notation better just rewrite the two lines without linking to primorial or a MathWorld reference, because they'd disagree, and n-th primorial  A002110(n) could be also clearer than primorial of n  (for a prime n) on OEIS proper.

Another Caldwell page is the first reference here, sadly both don't answer who (in a web search I found some results assuming that it's true, and "non-constructive", i.e., using the axiom of choice, thin ice, but that's only me :-) –Frank Ellermann 04:21, 22 October 2017 (UTC)