Brocard's problem
Does have integer solutions other than ?
Brocard's problem is a problem in mathematics that seeks integer values of such that is a perfect square, where is the factorial. Only three values of are known — 4, 5, 7 — and it is not known whether there are any more.
More formally, it seeks pairs of integers and such that
Brown numbers
Pairs of the numbers 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
References
- ^ Brocard, H. (1876), "Question 166", Nouv. Corres. Math., 2: 287
- ^ Brocard, H. (1885), "Question 1532", Nouv. Ann. Math., 4: 391
- MR 2280843
- ^ Pickover, Clifford A. (1995), Keys to Infinity, John Wiley & Sons, p. 170
- S2CID 119711158
- ^ 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
- MR 1204060
- MR 1430045
- MR 1951531
Further reading
- Guy, R. K. (2004), "D25: Equations involving factorial ", 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