1,1

Next term, if it exists, is greater than 10^850. - Sascha Kurz (sascha.kurz(AT)uni-bayreuth.de), Sep 22 2003

No more terms < 10^20000. - David Wasserman (wasserma(AT)spawar.navy.mil), Feb 08 2005

The problem of whether there are any other terms in this sequence, Brocard's problem, has been unsolved since 1876. It is virtually certain that there are no other terms and the known calculations give a(4) > (10^9)! = factorial(10^9). - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 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

Berndt and Galway, http://www.math.uiuc.edu/~berndt/articles/galway.pdf

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

5^2=25=4!+1

11^2=121=5!+1

71^2=5041=7!+1

Sequence in context: A025283 A076433 A069668 this_sequence A087399 A030081 A075047

Adjacent sequences: A085689 A085690 A085691 this_sequence A085693 A085694 A085695

nonn,bref

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