A146968 Brocard's problem: positive integers n such that n!+1 = m^2.

4, 5, 7

Offset: 1

Author

Marco Bellaccini (marcomurk(AT)tele2.it), Nov 03 2008

Keywords

• bref
• nonn
• hard

Comments

No other terms below 10^9.

See A085692 for more comments and references. - M. F. Hasler, Nov 20 2018

Examples

			7! + 1 = 5041 = 71^2, hence 7 is in the sequence. - _Klaus Brockhaus_, Nov 05 2008
		

Links

• Berndt, B. C. and Galway, W. F. On the Brocard-Ramanujan Diophantine Equation n!+1=m^2, The Ramanujan Journal, March 2000, Volume 4, Issue 1, pp 41-42.
• Apoloniusz Tyszka, On sets X subset of N for which we know an algorithm that computes a threshold number t(X) in N such that X is infinite if and only if X contains an element greater than t(X), 2019.
• Eric Weisstein's World of Mathematics, Brocard's Problem.

Crossrefs

A085692, A146968, A216071 are all essentially the same sequence. - N. J. A. Sloane, Sep 01 2012

Programs

Magma

[ : n in [1..8047] | t where t,p:=IsSquare(Factorial(n)+1) ]; // Klaus Brockhaus, Nov 05 2008

Mathematica

Select[Range[10],IntegerQ[Sqrt[#!+1]]&] (* Harvey P. Dale, Jan 31 2015 *)

PARI

{ for (n=1, 60100, if(issquare(n!+1) == 1, print(n) ) ) } \\ Marco Bellaccini (marcomurk(AT)tele2.it), Nov 08 2008

Shell

#!/bin/sh n=0 while(true) do n=`echo $n + 1 | bc` calc "($n! + 1)" ^ "(1 / 2)" | grep -v \. done

Extensions

Edited by Max Alekseyev, Feb 06 2010