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A085692 Brocard's problem: squares which can be written as n!+1 for some n.

%I A085692 #83 Jul 02 2025 16:55:22
%S A085692 25,121,5041
%N A085692 Brocard's problem: squares which can be written as n!+1 for some n.
%C A085692 Next term, if it exists, is greater than 10^850. - _Sascha Kurz_, Sep 22 2003
%C A085692 No more terms < 10^20000. - _David Wasserman_, Feb 08 2005
%C A085692 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 A085692 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 A085692 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 A085692 R. Guy, "Unsolved Problems in Number Theory", 3rd edition, D25
%D A085692 Clifford A. Pickover, A Passion for Mathematics (2005) at 69, 306.
%D A085692 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 19.
%H A085692 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 A085692 Robert D. Matson, <a href="http://unsolvedproblems.org/S73.pdf">Brocard's Problem 4th Solution Search Utilizing Quadratic Residues</a>, Unsolved Problems.
%H A085692 Wikipedia, <a href="https://en.wikipedia.org/wiki/Brocard%27s_problem">Brocard's problem</a>.
%F A085692 a(n) = A216071(n)^2 = A146968(n)!+1 = A038507(A146968(n)). - _M. F. Hasler_, Nov 20 2018
%e A085692    5^2 =   25 = 4! + 1;
%e A085692   11^2 =  121 = 5! + 1;
%e A085692   71^2 = 5041 = 7! + 1.
%t A085692 Select[Range[0,100]!+1,IntegerQ[Sqrt[#]] &] (* _Stefano Spezia_, Jul 02 2025 *)
%o A085692 (PARI) A085692=select( issquare, vector(99,n,n!+1)) \\ _M. F. Hasler_, Nov 20 2018
%Y A085692 A085692, A146968, A216071 are all essentially the same sequence. - _N. J. A. Sloane_, Sep 01 2012
%K A085692 nonn,bref
%O A085692 1,1
%A A085692 Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jul 18 2003