A321883 Nonnegative integers n for which n! + 1 is not a square.

0, 1, 2, 3, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69

Offset: 1

Author

Felix Fröhlich, Nov 20 2018

Keywords

• nonn

Comments

Complement of A146968 = positive integers n such that n!+1 is a square (Brocard's problem, so far {4, 5, 7} are the only known terms).

A weak form of Szpiro's conjecture implies that there are only finitely many nonnegative integers that are not in the sequence (cf. Overholt, 1993).

Links

• B. C. Berndt and W. F. Galway, On the Brocard-Ramanujan Diophantine Equation n! + 1 = m^2, The Ramanujan Journal, Vol. 4, No. 1 (2000), 41-42.
• M. Overholt, The Diophantine Equation n! + 1 = m^2, Bulletin of the London Mathematical Society, Vol. 25, No. 2 (1993), 104.
• Wikipedia, Brocard's problem

Crossrefs

Cf. A085692, A146968, A216071.

Programs

Mathematica

Select[Range[0,100], !IntegerQ[Sqrt[#!+1]] &] (* Amiram Eldar, Nov 21 2018 *)

PARI

select( is(n)=!issquare(n!+1), [0..99]) \\ M. F. Hasler, Nov 20 2018