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A085692 Brocard's problem: squares which can be written as n!+1 for some n. 10
25, 121, 5041 (list; graph; refs; listen; history; text; internal format)
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 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

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.uni-duesseldorf.de), Jul 18 2003

STATUS

approved

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Last modified July 2 02:41 EDT 2020. Contains 335389 sequences. (Running on oeis4.)