A088332 Primes of the form k! + 1.

2, 3, 7, 39916801, 10888869450418352160768000001, 13763753091226345046315979581580902400000001, 33452526613163807108170062053440751665152000000001

Offset: 1

Author

Cino Hilliard, Nov 06 2003

Keywords

• nonn

Comments

The next term is too large to include.

Of course 2 = 0! + 1 = 1! + 1 has two such representations.

Prime numbers that are the sum of two factorial numbers. - Juri-Stepan Gerasimov, Nov 08 2010

Examples

			3! + 1 = 7 is prime.
		

References

• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 118.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..15 (terms 1..11 from T. D. Noe)
• Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2023. - From _N. J. A. Sloane_, Jun 13 2012

Crossrefs

Cf. A002981 (values of k), A038507, A062701.

Programs

Mathematica

lst={};Do[p=n!+1;If[PrimeQ[p],AppendTo[lst,p]],{n,0,3*5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 27 2009 *)
Select[Range[50]!+1,PrimeQ] (* Harvey P. Dale, May 17 2025 *)

PARI

factp1prime(n)=for(x=1,n,xf=x!+1; if(isprime(xf),print1(xf",")))

Formula

a(n) = A038507(A002981(n+1)). - Elmo R. Oliveira, Apr 16 2025