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A085692 Brocard's problem: squares which can be written as n!+1 for some n. 10

%I

%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. It is virtually certain that there are no other terms and 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

%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 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

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Last modified October 23 22:28 EDT 2019. Contains 328373 sequences. (Running on oeis4.)