20210925, 22:50  #23  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{2}×2,393 Posts 
Quote:
The middle part primes (array of Y_{i}) are of course known,  and I do add a comment to them. That comment will make them live longer than they would given the rate the Top5000 grows. Those "Arithmetic progression (2,d=...)" primes are:As for the "Arithmetic progression (1,d=...)" primes, they owe their digital life to me: they were not eligible for Top5000 when they were found (they are vanity primes and were found exclusively to pump up "PrimeGrid credits"), and now that I made them members of the AP3 chain  they are. PrimeGrid is credited as a project that found them. ~Half of the seed array of X_{j} is actually in Top5000 in expired state; to those I will merely add a comment. 

20210926, 01:25  #24 
Feb 2017
Nowhere
5003_{10} Posts 
With a = 33*2^2939064  5606879602425*2^1290000  1 and d = 33*2^2939063  5606879602425*2^1290000, a == 2 (mod 5) and d == 4 (mod 5), so a + 2*d is divisible by 5. Thus a, a + d, a + 2d is not a 3term AP of primes.
5606879602425*2^1290000  1 No prime factors < 2^28. 33*2^2939063  1 prime (table lookup) 33*2^2939064  5606879602425*2^1290000  1 99*2^2939063  5606879602425*2^1290001  1 33*2^2939065  16820638807275*2^1290000  1 Divisible by 5 Estimates of baseten logs of a  d, a, a + d: (log(33)+2939063*log(2))/log(10) 884747.64066010744143595512237352714687 (log(33)+2939064*log(2))/log(10) 884747.94169010310541715033611242187136 (Matches value given here.) (log(99) + 2939063*log(2))/log(10) 884748.11778136216109839241740143040198 Thus a  d, a, a + d is (presumably) a 3term AP of primes. Luckily, a  d is an 884748 decimal digit number as well as a, so assuming this is the first term of an AP3 the digit count is still good. If a  2d = 5606879602425*2^1290000  1 happens to be prime, then a  2d, a  d, a, a + d is a 4term AP. But a  2d "only" has 388342 decimal digits (baseten log is 388341.44312776625375912068666134298687, approximately) 
20210926, 01:33  #25 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{2}×2,393 Posts 
I did test for the 4th AP term.

20210926, 14:02  #26  
Feb 2017
Nowhere
5,003 Posts 
Quote:
If the common difference is d, the AP3 is A_{3}  2d, A_{3}  d, A_{3}. I assume A_{4} was checked for small factors, then subjected to a compositeness test that proved it to be composite, but did not reveal any factors. 

20210929, 13:12  #27 
Feb 2017
Nowhere
1001110001011_{2} Posts 
In musing about APs of primes, I came upon a tantalizingly simple formula which, on the one hand very likely gives infinitely many longestpossible AP's starting with a given odd prime, but the other hand, is likely to be computationally useless for producing either AP's with large last term, or APs with large numbers of terms. Using the asymptotic formula in the BatemanHorn conjecture shows that in either case, the prospects of success are vanishingly small.
This may serve to illustrate how numerical data can fail to reflect what theory predicts. It may also serve to illustrate the level of artistry that goes into hunting for AP's of three or more terms which consist entirely of primes. Let k be a positive integer, p_{k} the k^{th} prime number, N = p_{k}#, the product of the first k prime numbers. Let f_{j}(x) = p_{k+1} + N*j*x, j = 1 to p_{k+1}  1. Then the p_{k+1}  1 polynomials f_{j}(x) satisfy the hypotheses of the BatemanHorn conjecture. (Note that with f_{0} = p_{k+1} however, f_{0} would be identically 0 (mod p_{k+1})) Thus, it would seem that for each positive integer k, there are infinitely many APp_{k+1}'s beginning with p_{k+1} and having common difference p_{k}#*x for positive integer x. The first few such (with the smallest x for which the AP's consist entirely of primes) are 3, 3 + 2x, 3 + 4x (x = 1) 5, 5 + 6x, 5 + 12x, 5 + 18x, 5 + 24x (x = 1) 7, 7 + 30x, 7 + 60x, 7 + 90x, 7 + 120x, 7 + 150x, 7 + 180x (x = 5) 11, 11 + 210x, 11 + 420x, 11 + 630x, 11 + 840x, 11 + 1050x, 11 + 1260x, 11 + 1470x, 11 + 1680x, 11 + 1890x, 11 + 2100x (x = 7315048) The asymptotic formula in the BatemanHorn conjecture indicates that for x in the vicinity of a large number X, the probability that the p_{k+1}  1 degree1 functions all yield prime values is where for each k c_{k} is a positive constant. The reader may verify that according to this estimate. in the vicinity of 10^884000, values of x for which 2x + 3 and 4x + 3 are both prime will be so thin on the ground, a simpleminded numerical sweep is very unlikely to find any. This is true a fortiori for the longer AP's. Likewise, as k increases, the smallest value of x for which p_{k+1}, p_{k+1} + x*p_{k}#, ..., p_{k+1} + (p_{k+1}  1)*x*p_{k}# are all prime, is likely to be too large for a simpleminded sweep to find it in a reasonable length of time. Last fiddled with by Dr Sardonicus on 20210929 at 15:57 Reason: fignix optsy 
20211015, 16:21  #28 
Jan 2007
Germany
2·191 Posts 
AP Update
Site is updated.
http://www.pzktupel.de/JensKruseAndersen/aprecords.htm Incorrect or news or..., let me know. Last fiddled with by Cybertronic on 20211015 at 16:22 
20211016, 10:43  #29 
Jan 2007
Germany
2×191 Posts 
Up to date
I hope , I have caught all APupdates in order of time.
http://www.pzktupel.de/JensKruseAnde...ds.htm#records This link is now valid for later updates. Norman Last fiddled with by Cybertronic on 20211016 at 10:43 
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