OFFSET
1,1
COMMENTS
Next term, if it exists, is greater than 10^850. - Sascha Kurz, Sep 22 2003
No more terms < 10^20000. - David Wasserman, Feb 08 2005
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
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
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
REFERENCES
R. Guy, "Unsolved Problems in Number Theory", 3rd edition, D25
Clifford A. Pickover, A Passion for Mathematics (2005) at 69, 306.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 19.
LINKS
Bruce C. Berndt and William F. Galway, On the Brocard-Ramanujan Diophantine Equation n! + 1 = m^2, The Ramanujan Journal, March 2000, Volume 4, Issue 1, pp 41-42.
Robert D. Matson, Brocard's Problem 4th Solution Search Utilizing Quadratic Residues, Unsolved Problems.
Wikipedia, Brocard's problem.
FORMULA
EXAMPLE
5^2 = 25 = 4! + 1;
11^2 = 121 = 5! + 1;
71^2 = 5041 = 7! + 1.
MATHEMATICA
Select[Range[0, 100]!+1, IntegerQ[Sqrt[#]] &] (* Stefano Spezia, Jul 02 2025 *)
PROG
(PARI) A085692=select( issquare, vector(99, n, n!+1)) \\ M. F. Hasler, Nov 20 2018
CROSSREFS
KEYWORD
nonn,bref
AUTHOR
Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jul 18 2003
STATUS
approved