%I #39 Sep 08 2022 08:45:38

%S 4,5,7

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

%C No other terms below 10^9.

%C See A085692 for more comments and references. - _M. F. Hasler_, Nov 20 2018

%H Berndt, B. C. and Galway, W. F. <a href="http://www.math.uiuc.edu/~berndt/articles/galway.pdf">On the Brocard-Ramanujan Diophantine Equation n!+1=m^2</a>, The Ramanujan Journal, March 2000, Volume 4, Issue 1, pp 41-42.

%H Apoloniusz Tyszka, <a href="https://philarchive.org/rec/TYSDAS">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)</a>, 2019.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BrocardsProblem.html">Brocard's Problem</a>.

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

%t Select[Range[10],IntegerQ[Sqrt[#!+1]]&] (* _Harvey P. Dale_, Jan 31 2015 *)

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

%o (Magma) [ <n, p>: n in [1..8047] | t where t,p:=IsSquare(Factorial(n)+1) ]; // _Klaus Brockhaus_, Nov 05 2008

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

%Y A085692, A146968, A216071 are all essentially the same sequence. - _N. J. A. Sloane_, Sep 01 2012

%K bref,nonn,hard

%O 1,1

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

%E Edited by _Max Alekseyev_, Feb 06 2010