Brocard's problem

Unsolved problem in mathematics:

Does ${\displaystyle n!+1=m^{2}}$ have integer solutions other than ${\displaystyle n=4,5,7}$?

Brocard's problem is a problem in mathematics that asks to find integer values of ${\displaystyle n}$ and ${\displaystyle m}$ for which

${\displaystyle n!+1=m^{2},}$
where ${\displaystyle n!}$ is the factorial. It was posed by Henri Brocard in a pair of articles in 1876 and 1885,[1][2] and independently in 1913 by Srinivasa Ramanujan.[3]

Brown numbers

Pairs of the numbers ${\displaystyle (n,m)}$ that solve Brocard's problem were named Brown numbers by Clifford A. Pickover in his 1995 book Keys to Infinity, after learning of the problem from Kevin S. Brown.[4] As of May 2021, there are only three known pairs of Brown numbers:

(4,5), (5,11), and (7,71),

based on the equalities

4! + 1 = 52 = 25,
5! + 1 = 112 = 121, and
7! + 1 = 712 = 5041.

Paul Erdős conjectured that no other solutions exist. Computational searches up to one quadrillion have found no further solutions.[5][6][7]

Connection to the abc conjecture

It would follow from the abc conjecture that there are only finitely many Brown numbers.[8] More generally, it would also follow from the abc conjecture that

${\displaystyle n!+A=k^{2}}$
has only finitely many solutions, for any given integer ${\displaystyle A}$,[9] and that
${\displaystyle n!=P(x)}$
has only finitely many integer solutions, for any given polynomial ${\displaystyle P(x)}$ of degree at least 2 with integer coefficients.[10]

References

1. ^ Brocard, H. (1876), "Question 166", Nouv. Corres. Math., 2: 287
2. ^ Brocard, H. (1885), "Question 1532", Nouv. Ann. Math., 4: 391
3. ^ Ramanujan, Srinivasa (2000), "Question 469", in Hardy, G. H.; Aiyar, P. V. Seshu; Wilson, B. M. (eds.), Collected papers of Srinivasa Ramanujan, Providence, Rhode Island: AMS Chelsea Publishing, p. 327, ISBN 0-8218-2076-1, MR 2280843
4. ^ Pickover, Clifford A. (1995), Keys to Infinity, John Wiley & Sons, p. 170
5. ^ Berndt, Bruce C.; Galway, William F. (2000), "On the Brocard–Ramanujan Diophantine equation n! + 1 = m2" (PDF), Ramanujan Journal, 4 (1): 41–42, doi:10.1023/A:1009873805276, MR 1754629, S2CID 119711158
6. ^ Matson, Robert (2017), "Brocard's Problem 4th Solution Search Utilizing Quadratic Residues" (PDF), Unsolved Problems in Number Theory, Logic and Cryptography, archived from the original (PDF) on 2018-10-06, retrieved 2017-05-07
7. ^ Epstein, Andrew; Glickman, Jacob (2020), C++ Brocard GitHub Repository
8. ^ Overholt, Marius (1993), "The Diophantine equation n! + 1 = m2", The Bulletin of the London Mathematical Society, 25 (2): 104, doi:10.1112/blms/25.2.104, MR 1204060
9. ^ Dąbrowski, Andrzej (1996), "On the Diophantine equation x! + A = y2", Nieuw Archief voor Wiskunde, 14 (3): 321–324, MR 1430045
10. ^ Luca, Florian (2002), "The Diophantine equation P(x) = n! and a result of M. Overholt" (PDF), Glasnik Matematički, 37(57) (2): 269–273, MR 1951531

• Guy, R. K. (2004), "D25: Equations involving factorial ${\displaystyle n}$", Unsolved Problems in Number Theory (3rd ed.), New York: Springer-Verlag, pp. 301–302