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 2012-08-22, 05:17 #45 LaurV Romulan Interpreter     Jun 2011 Thailand 2×3×31×47 Posts Using a bit different logic I confirm all the PRP values with <200 digits found up to now. Moreover, if we let apart the leading "3" and use only the digits in the fractional decimal expansion, that would modify the primes for 3 and 31: Code: (11:46:26) gp > get_primes_in_pi(0,100,1,1) Found 0 at position 32. Checking for prime ... Found: prp=2 Found 1 at position 1. Checking for prime ... Found: prp=14159 Found 2 at position 6. Checking for prime ... Found: prp=26535897932384626433832795028841971693993751058209 Found 3 at position 9. Checking for prime ... Found: prp=35897 Found 4 at position 2. Checking for prime ... Found: prp=41 Found 5 at position 4. Checking for prime ... Found: prp=59 Found 6 at position 7. Checking for prime ... Found: prp=653 Found 7 at position 13. Checking for prime ... Found: prp=79 Found 8 at position 11. Checking for prime ... Found: prp=89 Found 9 at position 5. Checking for prime ... Found: prp=9265358979323 Code: Found 30 at position 64. Checking for prime ... Found: prp=307 Found 31 at position 137. Checking for prime ... Found: prp=317 Found 32 at position 15. Checking for prime ... Found: prp=32384626433832795028841971693993751058209749445923078164062862089986280348253421 Found 33 at position 24. Checking for prime ... Found: prp=33832795028841971 Found 34 at position 86. Checking for prime ... Found: prp=348253 Found 35 at position 9. Checking for prime ... Found: prp=35897 Found 36 at position 285. Checking for prime ... Found: prp=3607 The last parameter is "only_non_trivial_primes" that is, extend the numbers if they are prime already, and the third parameter is "use only decimal expansion (ignore the leading 3)".
 2012-08-22, 05:54 #46 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 2×33×132 Posts For the leading zero, the following prime must be in octal*! :-) (this doesn't change the answer though, it's still "02") Also, I've revisited the larger PRPs and let the searches run for a while more and found a few more PRPs starting with the leftmost "62": 3490-, 7734-, 11111-, and 17155-digit (the last two are reportable to Lifchitz2) ______ *C convention. printf("%d\n", 052); will print 42
2012-08-22, 09:07   #47
LaurV
Romulan Interpreter

Jun 2011
Thailand

2×3×31×47 Posts

Quote:
 Originally Posted by Batalov For the leading zero, the following prime must be in octal! :-)

Joking apart, I just did a re-check for all thingies under 10k digits. With this occasion I found out that everybody completely missed 97. It was prime by itself in the "trivial" case, so it was not mentioned in post #9, and it was forgotten after the rules changed. My pari found a nice 821 digits beauty for it starting from position 12.

 2012-08-22, 09:20 #48 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 2×33×132 Posts It was not forgotten in post #32. PRPs under 1000 digits are too easy to even mention. (And Lifchitz site has a cutoff of 10000 digits.) Only 17 was slightly more challenging.
 2012-08-22, 09:50 #49 LaurV Romulan Interpreter     Jun 2011 Thailand 2·3·31·47 Posts Ah, ok then. I anyhow reported to FDB the PRPs for 54 and 73 (with 499 respective 446 digits) which were not reported, after I re-discovered them, together with the PRP for 97 in discussion.
 2012-08-22, 20:03 #50 davar55     May 2004 New York City 108616 Posts Were you doing a(20) and a(96) in parallel? So is length of a(20) already known > length of a(96), assuming it resolves finitely? Great work.
 2012-08-23, 06:59 #51 zhongbii   Aug 2012 1 Posts You can look in another way : Is the first N digit of pi (including 3)is a prime ? Have a look at this 3 31 314159 31415926535897932384626433832795028841 what is the next "PI-PRIME"? ps I'm poor in English ..... sorry
 2012-08-23, 18:51 #52 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 2·33·132 Posts Yes, this is the sequence A005042 (the extended version of the A060421 sequence). We've already discussed these above. I suspect that multiple people searched for larger members of this sequence (in other words, we shouldn't think that the search stopped at the 78073; E.W.W.'s mention of the upper search limit is 6 years old).
 2012-08-31, 14:04 #53 davar55     May 2004 New York City 423010 Posts The OP defined a single sequence, but somewhat loosely. There are really an infinite number of sequences fi with the OP defining f1. In that sequence, though it wasn't perfectly clear due to the calculations presented, the primes were intended to be represented by themselves (e.g. a(2) = 2 not the P50 that was found ). But the examples showed that the OPer was uncertain about that point. So f2 would be the sequence of primes starting at all the same places in pi but the SECOND prime found. Similarly for f3 and up. I think just the first two sequences would cover all that the OP intended, but finding the primes starting at ANY point in pi (as e.g. from the 3 prefix, which is represented in the oeis) will lead to a somewhat interesting sequence.
 2012-09-21, 18:35 #54 davar55     May 2004 New York City 2×32×5×47 Posts Considering the surprising (to me at least) length of some of the a(*) being discovered just up to 100, especially at 10, 20, 96, and 98, I think this sequence is interesting enough to beg another question: Just how random are the digitis of pi really? If we were to generate oher such "random" sequences (perhaps the digits of e as transcendental or sqrt 2 as merely irrational but non-patterned), seeing similar prime subsequence patterns might make this worthy of number theoretical study. In any case, as merely observor now, may I ask: Is iit very hard to prove the biggest PRPs prime? What's the L&L accreditor you referred to? Is a(20) still chugging away? Thanks for all your great work.
2012-09-21, 21:54   #55
CRGreathouse

Aug 2006

3·52·79 Posts

Quote:
 Originally Posted by Batalov E.W.W.'s mention of the upper search limit is 6 years old
He since increased it to 127,523 if I read this correctly:
http://mathworld.wolfram.com/IntegerSequencePrimes.html

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