A057468 Numbers k such that 3^k - 2^k is prime.

2, 3, 5, 17, 29, 31, 53, 59, 101, 277, 647, 1061, 2381, 2833, 3613, 3853, 3929, 5297, 7417, 90217, 122219, 173191, 256199, 336353, 485977, 591827, 1059503

Offset: 1

Author

Robert G. Wilson v, Sep 09 2000

Keywords

• nonn
• hard
• nice
• more

Comments

Some of the larger entries may only correspond to probable primes.

The 1137- and 1352-digit values associated with the terms 2381 and 2833 have been certified prime with Primo. - Rick L. Shepherd, Nov 12 2002

Or, numbers k such that A001047(k) is prime. - Zak Seidov, Sep 17 2006

3^k - 2^k were proved prime for k = 3613, 3853, 3929, 5297, 7417 with Primo. - David Harrison, Jun 08 2011

Links

• Henri Lifchitz and Renaud Lifchitz, PRP Records.
• R. Miles, Synchronization points and associated dynamical invariants, Trans. Amer. Math. Soc. 365 (2013), 5503-5524.
• Primality certificates for 3613 to 7417

Crossrefs

Cf. A058765, A000043 (Mersenne primes), A001047 (3^n-2^n).

Subset of A000040.

Programs

Mathematica

Select[Prime@ Range@ 941, PrimeQ[3^# - 2^#] &] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 and modified by Robert G. Wilson v, Mar 15 2017 *)
ParallelMap[ If[ PrimeQ[3^# - 2^#], #, Nothing] &, Prime@ Range@ 941] (* Robert G. Wilson v, Jun 28 2017 *)

PARI

select(p->ispseudoprime(3^n-2^n), primes(100)) \\ Charles R Greathouse IV, Feb 11 2011

Extensions

a(20) = 90217 found by Mike Oakes, Feb 23 2001

Terms a(21) = 122219, a(22) = 173191, a(23) = 256199 were found by Mike Oakes in 2003-2005. Corresponding numbers of decimal digits are 58314, 82634, 122238.

a(24) = 336353 found by Mike Oakes, Oct 15 2007. It corresponds to a probable prime with 160482 decimal digits.

a(25) = 485977 found by Mike Oakes, Sep 06 2009; it corresponds to a probable prime with 231870 digits. - Mike Oakes, Sep 08 2009

a(26) = 591827 found by Mike Oakes, Aug 25 2009; it corresponds to a probable prime with 282374 digits.

a(27) = 1059503 found by Mike Oakes, Apr 12 2012; it corresponds to a probable prime with 505512 digits. - Mike Oakes, Apr 14 2012