Primes in π

[davar55]

Originally Posted by davar55

Finding primes in the digits of pi has been done,
so here is yet another primes in pi puzzle.

For every positive integer n (in decimal) find
the first occurrence in pi of the digits of that integer,
then the first prime constructed from the subsequent digits of pi.

Here's what I had in mind:
__________________________________
1 --> 14159
2 --> 2 26535897932384626433 (*)
3 --> 3
4 --> 41
5 --> 59 (*)
6 --> 653
7 --> 79 (*)
8 --> 89
9 --> 9265358979323 (or 97 ??)
10 -> 102701 (or 1058209749...6531873029_<19128> ?) (PRP)
11 --> 11
12 --> 12847564823 (or 12848111...678925903_<211> ?)
...
20 --> (...more than 215000-digit...)
...
62 --> 3490-digit Prime (and three more PRPs)
80 --> 41938-digit PRP.
81 --> 4834-digit PRP.
84 --> 3057-digit PRP.
96 --> 140165-digit PRP.
98 --> 61303-digit PRP.
up to 100 (except 20): all primes/PRPs are less than 1000-digits or shown above
__________________________________

(*) of course 2, 5 and 7 are prime
(**) My calculator only checks for small factors, so * and ** may not actually be prime

I think the list carried to (at least) 100 will possibly contain some more interestingly large primes,
or will the prime-checking facility be tested only by going to 1000, or beyond?
_______________

P.S. I could restore the original values for 2 and 10, but ... they were composite. You can easily see the original in post #2 (SB)

In the OP, I had to enter the digits of pi by hand into my calculator, and let it test for primality.
That procedure was error-prone, hence the wrong values for 2 (extended from the true value of 2) amd 10.
But I hurried to get my partial results out, because I knew the forum would do the calculations over.

I think this will be an important sequence and jumping off point - we'll see.

[MattcAnderson]

3.14159 2654
And that's all I know. The primes
There are 2 3 5 7. And there are more.

Thank you very much for starting this thread

Regards

Matt

[davar55]

I've just noticed:

the series as computed (but not as in post#74) can be used to reconstruct the
digits of pi, as the first occurrence of the positive integers in pi exhaust pi as
the sequence of integers progresses. there is overlap, as in the first 3 (intended
to be the lead 3) and the first 31 (also from 3.1) and the first 314 (which can be
reconstructed from the 314th prime in the series (which starts with 314... ).

hence we should probably index the lead 3 as index 0 of the digits of pi.
this loses no primes, they just appear later in the series.

also, when we confirm the first prp of a(20), this reconstruction of pi from the
output would allow a double check of both the digits of pi and all the intermediate primes.

this assumes: For any positive integer written in decimal it is possible to construct a
prime number by concatenating some finite sequence of decimal digits at its end.

which is certainly true.

[davar55]

It's been a while. Any new results ?

Has a(20) been resolved with a prime, or probable prime,
or has anyone tried proving that a(20) never produces a prime ?

Any other big results below a(1000) ?

[davar55]

Any resolution of a(20)?

How long does each PRP check take at 450K?

[davar55]

Originally Posted by davar55

Any resolution of a(20)?
How long does each PRP check take at 450K?

Hate being a pest but I'm really into this result.
Can you say how high a(20) has been tested?

[davar55]

Originally Posted by davar55

Hate being a pest but I'm really into this result.
Can you say how high a(20) has been tested?

Is anyone here still working on this one?

[science_man_88]

Originally Posted by davar55

Is anyone here still working on this one?

first thing that comes to mind is pari and using say FaR or my attempt at parvecsearch ( see pari commands thread) but with alteration and depending on which gets used to look for all primes in a given range that start with that and searching for them in parallel or something like that.

[danaj]

Originally Posted by science_man_88

first thing that comes to mind is pari and using say FaR or my attempt at parvecsearch ( see pari commands thread) but with alteration and depending on which gets used to look for all primes in a given range that start with that and searching for them in parallel or something like that.

See post 64 for some timings once the size gets large. Pari is 10 times slower than PFGW once the sizes get very large. One still needs to run something like BPSW on the result (e.g. Perl/ntheory, Pari, or WraithX mpz_prp), but that ought to be faster than using Pari for each value.

[gd_barnes]

It seems there was still a fair amount of information that needed to be brought together for this. So I doublechecked all previous calculations in this thread for sequences 0 thru 200 to 50,000 digits. I chose to use the following rules:
1. Include the 3 at the beginning.
2. Extend sequences if they are prime.

I used a simple PFGW script that would do one sequence at a time. Sans the very first posting in this thread, I did not find any errors. The differences were the extension of the prime sequences, i.e. 199 becomes 1995611 instead of just 199. For sequences > 50,000 digits, I did not doublecheck beyond 50K digits. I used the latest large PRPs found or the most recent search depth that was available.

Attached are the results with number of digits shown for each sequence if it is > 10.

[davar55]

Originally Posted by gd_barnes

It seems there was still a fair amount of information that needed to be brought together for this. So I doublechecked all previous calculations in this thread for sequences 0 thru 200 to 50,000 digits. I chose to use the following rules:
1. Include the 3 at the beginning.
2. Extend sequences if they are prime.

I used a simple PFGW script that would do one sequence at a time. Sans the very first posting in this thread, I did not find any errors. The differences were the extension of the prime sequences, i.e. 199 becomes 1995611 instead of just 199. For sequences > 50,000 digits, I did not doublecheck beyond 50K digits. I used the latest large PRPs found or the most recent search depth that was available.

Attached are the results with number of digits shown for each sequence if it is > 10.

Your document looks very clean and presentable. Seen this way, I think your two rules are an improvement on the OP,
and if/when the prime sequence for 20 is found, this is the list that should be "published".