Brocard's problem asks to find integer values of n for which n! + 1 = m2, where n! is the factorial. It was posed by H. Brocard in a pair of articles in 1876 and 1885, and independently in 1913 by Ramanujan. Brown numbers Pairs of the numbers (m, n) that solve Brocard's problem are called Brown numbers. There are only three known pairs of Brown numbers: (4,5), (5,11), and (7,71). Paul Erdős conjectured that no other solutions exist. Most recently Berndt & Galway (2000) performed calculations for n up to 109 and found no further solutions. Variants of the problem Dabrowski (1996) has shown that it would follow from the abc conjecture that n! + A = k2 has only finitely many solutions, for any given integer A. References * Berndt, Bruce C. & Galway, William F. (2000), "The Brocard–Ramanujan diophantine equation n! + 1 = m2", The Ramanujan Journal 4: 41–42, <http://www.math.uiuc.edu/~berndt/articles/galway.pdf>. * Brocard, H. (1876), "Question 166", Nouv. Corres. Math. 2: 287. * Brocard, H. (1885), "Question 1532", Nouv. Ann. Math. 4: 391. * Dabrowski, A. (1996), "On the Diophantine Equation x! + A = y2", Nieuw Arch. Wisk. 14: 321–324. * Guy, R. K. (1994), "D25: Equations Involving Factorial", Unsolved Problems in Number Theory (2nd ed.), New York: Springer-Verlag, pp. 193–194. Links * Eric W. Weisstein, Brocard's Problem at MathWorld. * Eric W. Weisstein, Brown Numbers at MathWorld. Retrieved from "http://en.wikipedia.org/" |
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