A058013 Smallest prime p such that (n+1)^p - n^p is prime.

2, 2, 2, 3, 2, 2, 7, 2, 2, 3, 2, 17, 3, 2, 2, 5, 3, 2, 5, 2, 2, 229, 2, 3, 3, 2, 3, 3, 2, 2, 5, 3, 2, 3, 2, 2, 3, 3, 2, 7, 2, 3, 37, 2, 3, 5, 58543, 2, 3, 2, 2, 3, 2, 2, 3, 2, 5, 3, 4663, 54517, 17, 3, 2, 5, 2, 3, 3, 2, 2, 47, 61, 19, 23, 2, 2, 19, 7, 2, 7, 3, 2, 331, 2, 179, 5, 2, 5, 3, 2, 2

Offset

1,1

Comments

The terms a(47) and a(60) [were] unknown. The sequence continues at a(48): 2, 3, 2, 2, 3, 2, 2, 3, 2, 5, 3, 4663, a(60)=?, continued at a(61): 17, 3, 2, 5, 2, 3, 3, 2, 2, 47, 61, 19, 23, 2, 2, 19, 7, 2, 7, 3, 2, 331, 2, 179, 5, 2, 5, 3, 2, 2. - Hugo Pfoertner, Aug 27 2004

In September and November 2005, Jean-Louis Charton found a(60)=54517 and a(47)=58543, respectively. Earlier, Mike Oakes found a(106)=7639 and a(124)=5839. All these large values of a(n) yield probable primes. - T. D. Noe, Dec 05 2005, Sep 18 2008

a(106) = 6529 and a(124) = 5167 are true.

a(137) is probably 196873 from prime of this form discovered by Jean-Louis Charton in December 2009 and reported to Henri Lifchitz's PRP Top. - Robert Price, Feb 17 2012

a(138) through a(150) is 2,>32401,2,2,3,8839,5,7,2,3,5,271,13. - Robert Price, Feb 17 2012

a(276)=88301, a(139)>240000 and a(256)>100000. - Jean-Louis Charton, Jun 27 2012

Three more terms found, a(325)=81943, a(392)=64747, a(412)=56963 and also a(139)>260000, a(295)>100000, a(370)>100000, a(373)>100000. 29 unknown terms < 1000 remain. - Jean-Louis Charton, Aug 15 2012

Three more terms a(577)=55117, a(588)=60089 and a(756)=96487. - Jean-Louis Charton, Dec 13 2012

Three more (PRP) terms a(845)=83761, a(897)=48311, a(918)=54919. - Jean-Louis Charton, Dec 2012-2013.

Some of the results were computed using the PrimeFormGW (PFGW) primality-testing program. - Hugo Pfoertner, Nov 14 2019

Links

Robert Price and Robert G. Wilson v, Table of n, a(n) for n = 1..138

Jean-Louis Charton and Robert G. Wilson v, a(n) for n=1..1000 status

Richard Fischer, Generalized primes of the form (B+1)^N - B^N.

Formula

a((p-1)/2) = 2 for odd primes p. - Alexander Adamchuk, Dec 01 2006

Mathematica

lmt = 10000; f[n_] := Block[{p = 2}, While[p < lmt && !PrimeQ[(n+1)^p - n^p], p = NextPrime@ p]; If[p > lmt, 0, p]]; Do[ Print[{n, f[n] // Timing}], {n, 1000}] (* Robert G. Wilson v, Dec 01 2014 *)

Prog

(PARI) a(n)=forprime(p=2, default(primelimit), if(ispseudoprime((n+1)^p-n^p), return(p))) \\ Charles R Greathouse IV, Feb 20 2012

Crossrefs

Cf. A065913, A103794, A115596, A247244.

Cf. A000043, A057468, A059801, A059802, A062572-A062666, A215538.

Sequence in context: A325783 A048288 A050677 * A223934 A237531 A238504

Adjacent sequences: A058010 A058011 A058012 * A058014 A058015 A058016

Keyword

nonn,nice

Author

Robert G. Wilson v, Nov 13 2000

Extensions

More terms from T. D. Noe, Dec 05 2005

Typo in Mathematica program corrected by Ray Chandler, Feb 22 2017

Status

approved