A216071 -id:A216071 - cp's OEIS frontend

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

25, 121, 5041

Offset: 1

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

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

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

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.

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.

A085692, A146968, A216071 are all essentially the same sequence. - N. J. A. Sloane, Sep 01 2012

Select[Range[0,100]!+1,IntegerQ[Sqrt[#]] &] (* Stefano Spezia, Jul 02 2025 *)

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

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

A146968 Brocard's problem: positive integers n such that n!+1 = m^2.

Original entry on oeis.org

4, 5, 7

Offset: 1

Marco Bellaccini (marcomurk(AT)tele2.it), Nov 03 2008

No other terms below 10^9.

See A085692 for more comments and references. - M. F. Hasler, Nov 20 2018

			7! + 1 = 5041 = 71^2, hence 7 is in the sequence. - _Klaus Brockhaus_, Nov 05 2008

Berndt, B. C. and Galway, W. F. On the Brocard-Ramanujan Diophantine Equation n!+1=m^2, The Ramanujan Journal, March 2000, Volume 4, Issue 1, pp 41-42.
Apoloniusz Tyszka, On sets X subset of N for which we know an algorithm that computes a threshold number t(X) in N such that X is infinite if and only if X contains an element greater than t(X), 2019.
Eric Weisstein's World of Mathematics, Brocard's Problem.

A085692, A146968, A216071 are all essentially the same sequence. - N. J. A. Sloane, Sep 01 2012

[ : n in [1..8047] | t where t,p:=IsSquare(Factorial(n)+1) ]; // Klaus Brockhaus, Nov 05 2008

Select[Range[10],IntegerQ[Sqrt[#!+1]]&] (* Harvey P. Dale, Jan 31 2015 *)

{ for (n=1, 60100, if(issquare(n!+1) == 1, print(n) ) ) } \\ Marco Bellaccini (marcomurk(AT)tele2.it), Nov 08 2008

#!/bin/sh n=0 while(true) do n=`echo $n + 1 | bc` calc "($n! + 1)" ^ "(1 / 2)" | grep -v \. done

Edited by Max Alekseyev, Feb 06 2010

A232802 Number of solution pairs (x,y) for x <= 11 such that x! + n = y^2 (Brocard-Ramanujan Diophantine equation) is soluble over the integers.

Original entry on oeis.org

3, 1, 2, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 1, 1, 1, 0

Offset: 1

Frank M Jackson, Nov 30 2013

nonn

The Mathematica program will find the number of integer pairs (x,y) solving x!+n = y^2 for each n from 1 to 200 with x not exceeding 11. Dabrowski showed that the abc conjecture implies only finite solutions for each n. Berndt and Galway found that 11 was the highest value that x reached for a solution with n in the range 1 to 2500 and could find no further solution pairs (x,y) in that range even when x was increased to 10^5.

For n = 1 the number of solutions and arbitrary x is Brocard's problem, and it is conjectured - but verified only in the range x <= 10^12 - that there are 3 solution pairs (x,y): (4,5), (5,11), (7,71). - Georg Fischer, Nov 27 2020

Bruce Berndt and William Galway, On the Diophantine equation n! + 1 = m^2
Andrzej Dabrowski, On the Diophantine equation x! + A = y^2, Nieuw Archief voor Wiskunde, Vierde serie Deel 14 No. 3 (Nov. 1996) pp. 321-324.
Wikipedia, Brocard's problem
Wikipedia, abc conjecture - some consequences

Cf. A085692, A146968, A216071 (Brocard's problem; all essentially the same sequence).

Cf. A038202, A068869.

Table[Length@Select[Sqrt[Range[11]!+n], IntegerQ[#] &], {n, 1, 200}]

Definition narrowed by Georg Fischer, Nov 27 2020

A321883 Nonnegative integers n for which n! + 1 is not a square.

Original entry on oeis.org

0, 1, 2, 3, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69

Offset: 1

Felix Fröhlich, Nov 20 2018

nonn

Complement of A146968 = positive integers n such that n!+1 is a square (Brocard's problem, so far {4, 5, 7} are the only known terms).

A weak form of Szpiro's conjecture implies that there are only finitely many nonnegative integers that are not in the sequence (cf. Overholt, 1993).

B. C. Berndt and W. F. Galway, On the Brocard-Ramanujan Diophantine Equation n! + 1 = m^2, The Ramanujan Journal, Vol. 4, No. 1 (2000), 41-42.
M. Overholt, The Diophantine Equation n! + 1 = m^2, Bulletin of the London Mathematical Society, Vol. 25, No. 2 (1993), 104.
Wikipedia, Brocard's problem

Cf. A085692, A146968, A216071.

Select[Range[0,100], !IntegerQ[Sqrt[#!+1]] &] (* Amiram Eldar, Nov 21 2018 *)

select( is(n)=!issquare(n!+1), [0..99]) \\ M. F. Hasler, Nov 20 2018

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

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A146968 Brocard's problem: positive integers n such that n!+1 = m^2.

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A232802 Number of solution pairs (x,y) for x <= 11 such that x! + n = y^2 (Brocard-Ramanujan Diophantine equation) is soluble over the integers.

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A321883 Nonnegative integers n for which n! + 1 is not a square.

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