A047658 Numbers k such that the initial k digits in decimal portion of Pi form a prime number.

5, 12, 281, 547, 6205, 16350

Offset: 1

Author

Carlos Rivera

Keywords

• hard
• nice
• nonn
• base
• more

Comments

Conjecture: this sequence is finite. - Carlos Rivera

Rivera's conjecture that this sequence is finite conflicts with heuristics; the next entry is almost certainly 6205, since floor((Pi-3)*10^6205) is (very) probably prime, though its proof may take decades. - David Broadhurst, Nov 08 2000

Floor((Pi-3)*10^6205) is a strong pseudoprime to all (1229) prime bases a < 10000 (the test took 45 minutes). - Joerg Arndt, Jan 16 2011

Terms for n>=5 are only probable primes. - Dmitry Kamenetsky, Aug 03 2015

Floor((Pi-3)*10^16350) is a probable prime, checked with 25 iterations of the Miller-Rabin test. - Dmitry Kamenetsky, Aug 05 2015

The next term is greater than 65400. - Dmitry Kamenetsky, Aug 09 2015

The next term is greater than 100000. - Michael S. Branicky, Sep 29 2024

Examples

			5 gives 14159 (prime); 12 gives 141592653589 (prime) and so on.
		

Links

• Chris K. Caldwell, Prime Curios: 14159...07021 (547-digits)
• Shyam Sunder Gupta, Mystery of pi, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 19, 473-497.

Crossrefs

Cf. A000796 (Pi), A060421, A064118.

Programs

Mathematica

nn=1000; d=RealDigits[Pi-3, 10, nn][[1]]; Select[Range[nn], PrimeQ[FromDigits[Take[d, #]]] &]

PARI

is(n)=isprime((Pi-3)*10^n\1) \\ Charles R Greathouse IV, Aug 28 2015

Python

from sympy import S, isprime
pi_digits = str(S.Pi.n(10**5))[2:-1]
def afind():
    kint = 0
    for k in range(len(pi_digits)):
        kint = 10*kint + int(pi_digits[k])
        if isprime(kint):
            print(k+1, end=", ")
afind() # Michael S. Branicky, Jan 29 2023