Brocard's problem

Unsolved problem in mathematics:

Does

n!+1=m^{2}

have integer solutions other than

n=4,5,7

?

(more unsolved problems in mathematics)

Brocard's problem is a problem in mathematics that seeks integer values of $n$ such that $n!+1$ is a perfect square, where $n!$ is the factorial. Only three values of $n$ are known — 4, 5, 7 — and it is not known whether there are any more.

More formally, it seeks pairs of integers $n$ and $m$ such that $n!+1=m^{2}.$ The problem 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 $(n,m)$ 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 October 2022, 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.^[5] Computational searches up to one quadrillion have found no further solutions.^[6]^[7]^[8]

Connection to the abc conjecture

It would follow from the abc conjecture that there are only finitely many Brown numbers.^[9] More generally, it would also follow from the abc conjecture that $n!+A=k^{2}$ has only finitely many solutions, for any given integer $A$ ,^[10] and that $n!=P(x)$ has only finitely many integer solutions, for any given polynomial $P(x)$ of degree at least 2 with integer coefficients.^[11]

References

^ Brocard, H. (1876), "Question 166", Nouv. Corres. Math., 2: 287
^ Brocard, H. (1885), "Question 1532", Nouv. Ann. Math., 4: 391
^ 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
^ Pickover, Clifford A. (1995), Keys to Infinity, John Wiley & Sons, p. 170
^ Erdős, Paul (1963), "Quelques problèmes de la théorie des nombres" (PDF), in Chabauty, C.; Chatelet, A.; Chatelet, F.; Descombes, R.; Pisot, C.; Poitou, G. (eds.), Introduction à la théorie des nombres, Monographies de l'Enseignement Mathématique (in French), vol. 6, University of Geneva, pp. 81–135; see problème 67, p. 129
^ Berndt, Bruce C.; Galway, William F. (2000), "On the Brocard–Ramanujan Diophantine equation $n! + 1 = m 2$ " (PDF), Ramanujan Journal, 4 (1): 41–42, doi:10.1023/A:1009873805276, MR 1754629, S2CID 119711158, archived from the original (PDF) on 2017-07-03
^ 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
^ Epstein, Andrew; Glickman, Jacob (2020), C++ Brocard GitHub Repository
^ Overholt, Marius (1993), "The Diophantine equation $n! + 1 = m 2$ ", The Bulletin of the London Mathematical Society, 25 (2): 104, doi:10.1112/blms/25.2.104, MR 1204060
^ Dąbrowski, Andrzej (1996), "On the Diophantine equation $x! + A = y 2$ ", Nieuw Archief voor Wiskunde, 14 (3): 321–324, MR 1430045
^ 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

External links

Weisstein, Eric W., "Brocard's Problem" ("Brown Numbers") at MathWorld.
Copeland, Ed, "Brown Numbers", Numberphile, Brady Haran, archived from the original on 2014-11-09, retrieved 2013-04-06

[broc1-1] Brocard, H. (1876), "Question 166", Nouv. Corres. Math., 2: 287

[broc2-2] Brocard, H. (1885), "Question 1532", Nouv. Ann. Math., 4: 391

[ramanujan-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

[pickover-4] Pickover, Clifford A. (1995), Keys to Infinity, John Wiley & Sons, p. 170

[erdos-5] Erdős, Paul (1963), "Quelques problèmes de la théorie des nombres" (PDF), in Chabauty, C.; Chatelet, A.; Chatelet, F.; Descombes, R.; Pisot, C.; Poitou, G. (eds.), Introduction à la théorie des nombres, Monographies de l'Enseignement Mathématique (in French), vol. 6, University of Geneva, pp. 81–135; see problème 67, p. 129

[bergal-6] Berndt, Bruce C.; Galway, William F. (2000), "On the Brocard–Ramanujan Diophantine equation $n! + 1 = m 2$ " (PDF), Ramanujan Journal, 4 (1): 41–42, doi:10.1023/A:1009873805276, MR 1754629, S2CID 119711158, archived from the original (PDF) on 2017-07-03

[matson-7] 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

[epsgli-8] Epstein, Andrew; Glickman, Jacob (2020), C++ Brocard GitHub Repository

[overholt-9] Overholt, Marius (1993), "The Diophantine equation $n! + 1 = m 2$ ", The Bulletin of the London Mathematical Society, 25 (2): 104, doi:10.1112/blms/25.2.104, MR 1204060

[dabrowski-10] Dąbrowski, Andrzej (1996), "On the Diophantine equation $x! + A = y 2$ ", Nieuw Archief voor Wiskunde, 14 (3): 321–324, MR 1430045

[luca-11] 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

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