%I #79 Sep 21 2022 05:18:14

%S 25,121,5041

%N Brocard's problem: squares which can be written as n!+1 for some n.

%C Next term, if it exists, is greater than 10^850. - _Sascha Kurz_, Sep 22 2003

%C No more terms < 10^20000. - _David Wasserman_, Feb 08 2005

%C The problem of whether there are any other terms in this sequence, Brocard's problem, has been unsolved since 1876. The known calculations give a(4) > (10^9)! = factorial(10^9). - _Stefan Steinerberger_, Mar 19 2006

%C I wrote a similar program sieving against the 40 smallest primes larger than 4*10^9 and can report that a(4) > factorial(4*10^9+1). In other words, it's now known that the only n <= 4*10^9 for which n!+1 is a square are 4, 5 and 7. C source code available on request. - Tim Peters (tim.one(AT)comcast.net), Jul 02 2006

%C Robert Matson claims to have verified that 4, 5, and 7 are the only values of n <= 10^12 for which n!+1 is a square. This implies that the next term, if it exists, is greater than (10^12+1)! ~ 1.4*10^11565705518115. - _David Radcliffe_, Oct 28 2019

%D R. Guy, "Unsolved Problems in Number Theory", 3rd edition, D25

%D Clifford A. Pickover, A Passion for Mathematics (2005) at 69, 306.

%H Bruce C. Berndt and William F. Galway, <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 Robert D. Matson, <a href="http://unsolvedproblems.org/S73.pdf">Brocard's Problem 4th Solution Search Utilizing Quadratic Residues</a>, Unsolved Problems.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Brocard%27s_problem">Brocard's problem</a>

%F a(n) = A216071(n)^2 = A146968(n)!+1 = A038507(A146968(n)). - _M. F. Hasler_, Nov 20 2018

%e 5^2 = 25 = 4! + 1;

%e 11^2 = 121 = 5! + 1;

%e 71^2 = 5041 = 7! + 1.

%o (PARI) A085692=select( issquare, vector(99,n,n!+1)) \\ _M. F. Hasler_, Nov 20 2018

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

%K nonn,bref

%O 1,1

%A Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jul 18 2003