Brocard's problem (original) (raw)

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In mathematics, when is n!+1 a square

Unsolved problem in mathematics:

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

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

More formally, it seeks pairs of integers n {\displaystyle n} {\displaystyle n} and m {\displaystyle m} {\displaystyle m} such that n ! + 1 = m 2 . {\displaystyle n!+1=m^{2}.} {\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]

Pairs of the numbers ( n , m ) {\displaystyle (n,m)} {\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.[5] Computational searches up to one quadrillion have found no further solutions.[6][7][8]

Connection to the abc conjecture

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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 {\displaystyle n!+A=k^{2}} {\displaystyle n!+A=k^{2}}has only finitely many solutions, for any given integer A {\displaystyle A} {\displaystyle A},[10] and that n ! = P ( x ) {\displaystyle n!=P(x)} {\displaystyle n!=P(x)}has only finitely many integer solutions, for any given polynomial P ( x ) {\displaystyle P(x)} {\displaystyle P(x)} of degree at least 2 with integer coefficients.[11]

  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. ^ 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
  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
  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
  8. ^ Epstein, Andrew; Glickman, Jacob (2020), C++ Brocard GitHub Repository
  9. ^ 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
  10. ^ Dąbrowski, Andrzej (1996), "On the Diophantine equation x! + A = _y_2", Nieuw Archief voor Wiskunde, 14 (3): 321–324, MR 1430045
  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