Английская Википедия:Brocard's problem

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Шаблон:Short description Шаблон:Distinguish Шаблон:Unsolved Brocard's problem is a problem in mathematics that seeks integer values of <math>n</math> such that <math>n!+1</math> is a perfect square, where <math>n!</math> is the factorial. Only three values of <math>n</math> are known — 4, 5, 7 — and it is not known whether there are any more.

More formally, it seeks pairs of integers <math>n</math> and <math>m</math> such that<math display="block">n!+1 = m^2.</math>The problem was posed by Henri Brocard in a pair of articles in 1876 and 1885,Шаблон:R and independently in 1913 by Srinivasa Ramanujan.Шаблон:R

Brown numbers

Pairs of the numbers <math>(n,m)</math> 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.Шаблон:R As of October 2022, there are only three known pairs of Brown numbers: Шаблон:Bi based on the equalities Шаблон:Bi Шаблон:Bi Шаблон:Bi Paul Erdős conjectured that no other solutions exist. Computational searches up to one quadrillion have found no further solutions.Шаблон:R

Connection to the abc conjecture

It would follow from the abc conjecture that there are only finitely many Brown numbers.Шаблон:R More generally, it would also follow from the abc conjecture that <math display=block>n!+A = k^2</math> has only finitely many solutions, for any given integer <math>A</math>,Шаблон:R and that <math display=block>n! = P(x)</math> has only finitely many integer solutions, for any given polynomial <math>P(x)</math> of degree at least 2 with integer coefficients.Шаблон:R

Further reading

• Шаблон:Citation

External links

• Шаблон:Mathworld2
• Шаблон:Citation