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
[COLOR=Red]Found 3 at position 9. Checking for prime ... Found: prp=35897[/COLOR]
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]

[CODE]Found 30 at position 64. Checking for prime ... Found: prp=307
[COLOR=Red]Found 31 at position 137. Checking for prime ... Found: prp=317
[/COLOR]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[/CODE]

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)".

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 Lifchitz[SUP]2[/SUP])

______
*C convention. printf("%d\n", 052); will print 42

[QUOTE=Batalov;308875]For the leading zero, the following prime must be in octal! :-)
[/QUOTE]
:razz:

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 [URL="http://factordb.com/index.php?id=1100000000530494297"]nice 821 digits[/URL] beauty for it starting from position 12.

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.

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.

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.

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

Yes, this is the sequence [URL="http://oeis.org/A005042"]A005042[/URL]
(the extended version of the [URL="http://oeis.org/A060421"]A060421[/URL] 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 [URL="http://mathworld.wolfram.com/Pi-Prime.html"]mention[/URL] of the upper search limit is 6 years old).

The OP defined a single sequence, but somewhat loosely.
There are really an infinite number of sequences f[sub]i[/sub]
with the OP defining f[sub]1[/sub].
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[sub]2[/sub] would be the sequence of primes starting at all the
same places in pi but the SECOND prime found. Similarly for f[sub]3[/sub]
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.

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.

[QUOTE=Batalov;309040]E.W.W.'s [URL="http://mathworld.wolfram.com/Pi-Prime.html"]mention[/URL] of the upper search limit is 6 years old[/QUOTE]

He since increased it to 127,523 if I read this correctly:
[url]http://mathworld.wolfram.com/IntegerSequencePrimes.html[/url]