20220204, 15:54  #1 
Jun 2003
Suva, Fiji
2^{3}×3×5×17 Posts 
Primes in order in pi
I am a little surprised I can't find any mention on the web, of the sequence of primes, in order, that might be found in the decimal expansion of pi (3.1415926535..)
The first instance of 2 is at the 6th decimal position, the first 3 thereafter at the 9th position, the first 5 thereafter at the 10th, etc. For definition's sake, the first 71 is not at the 1757th position following 67 at the 1756th, as the 1757th position was also used to ascertain 67. Instead, 71 is found at the 1923rd. A file of the decimal positions of all primes up to 8713 which are to be found in order in the first 10,000,000 decimal positions of pi is attached. A graph is also shown, showing the positions of each prime, supporting the hypothesis that pi is normal. Some of the results are shown below Code:
14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612847564823378678316527... Code:
#!/usr/bin/env perl use warnings; use strict; use Math::Prime::Util qw/:all/; use feature ':5.10'; $=1; my $pie = substr(Pi(10_000_000),2); my $maxprime = 100_000; my $prevplace = 0; forprimes { my $i = 1+index($pie,$_,$prevplace); say $_, " ", $i; $prevplace = $i+length($_)1; if ( $i==0){ last; } } 2,$maxprime; 
20220204, 16:29  #2 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
2^{4}×3^{3}×5^{2} Posts 
That inflection point around 1000 looks interesting. Any insight into that?

20220204, 17:04  #3  
Jun 2003
Suva, Fiji
11111111000_{2} Posts 
Quote:
When I inferred that pi is normal in the OP, I meant pi behaves as a normal number. Last fiddled with by robert44444uk on 20220204 at 17:48 

20220205, 01:48  #4 
Feb 2019
2^{4} Posts 
Here is my result for 100b digits of Pi.
All primes up to 1042333 were found in order in these 100b digits. 
20220205, 14:50  #5 
Feb 2017
Nowhere
17CE_{16} Posts 
Assume pi is a normal number to base ten.
It occurred to me to wonder how many times the block of decimal digits representing a prime p can reasonably be expected to occur before it occurs in its proper place in the sequence of primes. 
20220205, 14:59  #6 
"Oliver"
Sep 2017
Porta Westfalica, DE
4C5_{16} Posts 

20220205, 17:05  #7  
Feb 2019
20_{8} Posts 
Quote:
Here is the source code Code:
import java.io.*; public class PrimesInPi { public static void main(String[] args) throws IOException { FileReader fr = new FileReader("pi.txt"); long piSize = 100000000000L; final int maxArraySize = 1000000000; int chunkSize = (piSize>maxArraySize) ? maxArraySize : (int) piSize; char[][] pi; pi = new char[(int) (piSize/chunkSize)][chunkSize]; fr.read(); fr.read(); for (int i=0; i<piSize/chunkSize; i++) { fr.read(pi[i], 0, chunkSize); } fr.close(); BufferedReader primesReader = new BufferedReader(new FileReader("primes.txt")); String prime = primesReader.readLine(); FileWriter result = new FileWriter("output.txt"); long pos = 0; int i=0, j=0; while (pos<piSize) { if (isCurrentPrime(pi, prime, i, j)) { result.write(prime+" "+(pos+1)+"\n"); prime = primesReader.readLine(); j+=prime.length(); pos+=prime.length(); if (j>=chunkSize) { j=chunkSize; i++; } } else { pos++; j++; if (j==chunkSize) { j=0; i++; } } } primesReader.close(); result.close(); } private static boolean isCurrentPrime(char[][] pi, String prime, int i, int j) { boolean result = true; int i1 = i, j1 = j; for (int k=0; k<prime.length(); k++) { if (prime.charAt(k) == pi[i1][j1]) { j1++; if (j1 == pi[i1].length) { j1 = 0; i1++; } if (i1 == pi.length) { result = false; break; } } else { result = false; break; } } return result; } } piSize variable should set to count of pi digits which we want to process. 

20220205, 17:16  #8 
"Oliver"
Sep 2017
Porta Westfalica, DE
1221_{10} Posts 
Thank you!
For some reason I was expecting that the straightforward approach would be inefficient but you are correct that this approach can be applied to a file that is not fully stored in memory. I'll code up something later. I like your idea to have the primes precalculated in a file. This makes the code way less complex. 
20220205, 17:59  #9  
Jan 2017
98_{16} Posts 
Quote:
So the expected number should be (number of earlier equally long target sequences) + (number of one earlier one digit shorter target sequences)/10 + (two shorter)/100 and so on. 

20220207, 09:15  #10 
Jun 2003
Suva, Fiji
2^{3}·3·5·17 Posts 
You have to love the projected result of this exercise  there is exists within the decimal representation of pi, (which is itself just one of an infinite number of irrationals), all of the infinity of primes, in order.
I would expect that this projected result will be the same for all integer bases, i.e. the representations of the primes in any integer base can be found in order within the representation of pi in that base. I can almost visualise this must be true for base 2. Can we go further to project or hypothesise that the primes, as depicted in any integer base are in order in the representation of any irrational number in the same integer base? As integers represented in irrational bases appear themselves irrational, then we can safely say that the primes in order, represented in irrational base form, cannot be found in the representation of the irrational in that base, as the irrational in its own base will have a simple nonirrational form. Because the representation of the irrational form is infinite, I would hypothesise that the primes represented in irrational base form (which I think are irrational representations themselves) are not included in order in the representation of an irrational number presented in an different rational or irrational base. Have I covered all the permutations here? PS: My teacher at school did day "don't mess with infinity, it bites" 
20220207, 12:11  #11  
Jan 2017
2^{3}×19 Posts 
Quote:
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