Brocard's problem

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Short description: In mathematics, when is n!+1 a square
Question, Web Fundamentals.svg 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)

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:

(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 [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

  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., 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 
  4. Keys to Infinity, John Wiley & Sons, 1995, p. 170 
  5. 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 
  6. 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 
  7. Epstein, Andrew; Glickman, Jacob (2020), C++ Brocard GitHub Repository, https://github.com/jhg023/brocard 
  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 
  9. Dąbrowski, Andrzej (1996), "On the Diophantine equation x! + A = y2", Nieuw Archief voor Wiskunde 14 (3): 321–324 
  10. 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