Brocard's problem
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Unsolved problem in mathematics: Does [math]\displaystyle{ n!+1=m^2 }[/math] have integer solutions other than [math]\displaystyle{ n=4,5,7 }[/math]? (more unsolved problems in mathematics)
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Brocard's problem is a problem in mathematics that seeks integer values of [math]\displaystyle{ n }[/math] such that [math]\displaystyle{ n!+1 }[/math] is a perfect square, where [math]\displaystyle{ n! }[/math] is the factorial. Only three values of [math]\displaystyle{ n }[/math] are known — 4, 5, 7 — and it is not known whether there are any more.
More formally, it seeks pairs of integers [math]\displaystyle{ n }[/math] and [math]\displaystyle{ m }[/math] such that[math]\displaystyle{ n!+1 = m^2. }[/math]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 [math]\displaystyle{ (n,m) }[/math] 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:
based on the equalities
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 [math]\displaystyle{ n!+A = k^2 }[/math] has only finitely many solutions, for any given integer [math]\displaystyle{ A }[/math],[9] and that [math]\displaystyle{ n! = P(x) }[/math] has only finitely many integer solutions, for any given polynomial [math]\displaystyle{ P(x) }[/math] of degree at least 2 with integer coefficients.[10]
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., Collected papers of Srinivasa Ramanujan, Providence, Rhode Island: AMS Chelsea Publishing, p. 327, ISBN 0-8218-2076-1, https://books.google.com/books?id=h1G2CgAAQBAJ&pg=PA327
- ↑ Keys to Infinity, John Wiley & Sons, 1995, p. 170
- ↑ Berndt, Bruce C.; Galway, William F. (2000), "On the Brocard–Ramanujan Diophantine equation n! + 1 = m2", Ramanujan Journal 4 (1): 41–42, doi:10.1023/A:1009873805276, https://www.math.uiuc.edu/~berndt/articles/galway.pdf
- ↑ Matson, Robert (2017), "Brocard's Problem 4th Solution Search Utilizing Quadratic Residues", Unsolved Problems in Number Theory, Logic and Cryptography, http://unsolvedproblems.org/S99.pdf, retrieved 2017-05-07
- ↑ Epstein, Andrew; Glickman, Jacob (2020), C++ Brocard GitHub Repository, https://github.com/jhg023/brocard
- ↑ 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
- ↑ Dąbrowski, Andrzej (1996), "On the Diophantine equation x! + A = y2", Nieuw Archief voor Wiskunde 14 (3): 321–324
- ↑ Luca, Florian (2002), "The Diophantine equation P(x) = n! and a result of M. Overholt", Glasnik Matematički 37(57) (2): 269–273, https://web.math.hr/glasnik/37.2/37(2)-04.pdf
Further reading
- "D25: Equations involving factorial [math]\displaystyle{ n }[/math]", Unsolved Problems in Number Theory (3rd ed.), New York: Springer-Verlag, 2004, pp. 301–302
External links
- Eric W. Weisstein, Brocard's Problem (Brown Numbers) at MathWorld.
- Copeland, Ed, "Brown Numbers", Numberphile (Brady Haran), http://www.numberphile.com/videos/brown_numbers.html, retrieved 2013-04-06
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