

A085692


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


10




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. 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
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.


LINKS

Table of n, a(n) for n=1..3.
Bruce C. Berndt and William F. Galway, On the BrocardRamanujan Diophantine Equation n! + 1 = m^2, The Ramanujan Journal, March 2000, Volume 4, Issue 1, pp 4142.
Robert D. Matson, Brocard's Problem 4th Solution Search Utilizing Quadratic Residues, Unsolved Problems.
Wikipedia, Brocard's problem


FORMULA

a(n) = A216071(n)^2 = A146968(n)!+1 = A038507(A146968(n)).  M. F. Hasler, Nov 20 2018


EXAMPLE

5^2 = 25 = 4! + 1;
11^2 = 121 = 5! + 1;
71^2 = 5041 = 7! + 1.


PROG

(PARI) A085692=select( issquare, vector(99, n, n!+1)) \\ M. F. Hasler, Nov 20 2018


CROSSREFS

A085692, A146968, A216071 are all essentially the same sequence.  N. J. A. Sloane, Sep 01 2012
Sequence in context: A069668 A274785 A214114 * A087399 A030081 A075047
Adjacent sequences: A085689 A085690 A085691 * A085693 A085694 A085695


KEYWORD

nonn,bref


AUTHOR

Klaus Strassburger (strass(AT)ddfi.uniduesseldorf.de), Jul 18 2003


STATUS

approved



