20120822, 05:17  #45 
Romulan Interpreter
Jun 2011
Thailand
2^{4}·13·47 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 
20120822, 05:54  #46 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
22534_{8} 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 17155digit (the last two are reportable to Lifchitz^{2}) ______ *C convention. printf("%d\n", 052); will print 42 
20120822, 09:07  #47 
Romulan Interpreter
Jun 2011
Thailand
2^{4}×13×47 Posts 
Joking apart, I just did a recheck 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. 
20120822, 09:20  #48 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
9564_{10} 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. 
20120822, 09:50  #49 
Romulan Interpreter
Jun 2011
Thailand
2630_{16} 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 rediscovered them, together with the PRP for 97 in discussion. 
20120822, 20:03  #50 
May 2004
New York City
10213_{8} 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. 
20120823, 06:59  #51 
Aug 2012
1_{2} 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 "PIPRIME"? ps I'm poor in English ..... sorry 
20120823, 18:51  #52 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{2}×3×797 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). 
20120831, 14:04  #53 
May 2004
New York City
10213_{8} Posts 
The OP defined a single sequence, but somewhat loosely.
There are really an infinite number of sequences f_{i} with the OP defining f_{1}. 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 f_{2} would be the sequence of primes starting at all the same places in pi but the SECOND prime found. Similarly for f_{3} 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. 
20120921, 18:35  #54 
May 2004
New York City
5·7·11^{2} 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 nonpatterned), 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. 
20120921, 21:54  #55  
Aug 2006
13533_{8} Posts 
Quote:
http://mathworld.wolfram.com/IntegerSequencePrimes.html 

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