Brocard's problem asks to find integer values of n for which

n! + 1 = m²,

where n! is the factorial. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French Definition The factorial function is formally defined by n!=\prod_{k=1}^n k 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

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. Paul Erdős ( Hungarian: Erdős Pál, in English occasionally Paul Erdos or Paul Erdös, March 26, 1913 &ndash 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. The abc conjecture is a Conjecture in Number theory. It was first proposed by Joseph Oesterlé and David Masser in 1985

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. Richard Kenneth Guy (born 1916 Nuneaton, Warwickshire) is a British mathematician Professor Emeritus in the Department of Mathematics ), New York: Springer-Verlag, pp. 193–194 .

External links

• Eric W. Weisstein, Brocard's Problem at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
• Eric W. Weisstein, Brown Numbers at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W

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