A047658 - OEIS

%I #59 Sep 30 2024 01:19:30

%S 5,12,281,547,6205,16350

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

%C Conjecture: this sequence is finite. - _Carlos Rivera_

%C 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

%C 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

%C Terms for n>=5 are only probable primes. - _Dmitry Kamenetsky_, Aug 03 2015

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

%C The next term is greater than 65400. - _Dmitry Kamenetsky_, Aug 09 2015

%C The next term is greater than 100000. - _Michael S. Branicky_, Sep 29 2024

%H C. K. Caldwell, <a href="https://t5k.org/curios/page.php?number_id=551">Prime Curios: 14159...07021 (547-digits)</a>

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

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

%o (PARI) is(n)=isprime((Pi-3)*10^n\1) \\ _Charles R Greathouse IV_, Aug 28 2015

%o (Python)

%o from sympy import S, isprime

%o pi_digits = str(S.Pi.n(10**5))[2:-1]

%o def afind():

%o     kint = 0

%o     for k in range(len(pi_digits)):

%o         kint = 10*kint + int(pi_digits[k])

%o         if isprime(kint):

%o             print(k+1, end=", ")

%o afind() # _Michael S. Branicky_, Jan 29 2023

%Y Cf. A000796 (Pi), A060421, A064118.

%K hard,nice,nonn,base,more

%O 1,1

%A _Carlos Rivera_