Primes in π

[LaurV]

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:

(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

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)".

[Batalov]

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

[LaurV]

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.

[Batalov]

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.

[LaurV]

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.

[davar55]

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.

[zhongbii]

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

[Batalov]

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).

[davar55]

The OP defined a single sequence, but somewhat loosely.
There are really an infinite number of sequences f_i
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.

[davar55]

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.

[CRGreathouse]

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