Brocard's problem

Not to be confused with Brocard's conjecture.

Unsolved problem in mathematics:

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

(more unsolved problems in mathematics)

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

More formally, it seeks pairs of integers ${\displaystyle n}$ and ${\displaystyle m}$ such that

$${\displaystyle n!+1=m^{2}.}$$

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 ${\displaystyle (n,m)}$ 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 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. 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

Further reading

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

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