Brocard's problem

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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.

Contents 1 Brown numbers 2 Variants of the problem 3 References 4 External links

[edit] 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.

[edit] 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.

[edit] 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 .

[edit] External links

• Eric W. Weisstein, Brocard's Problem at MathWorld.
• Eric W. Weisstein, Brown Numbers at MathWorld.