Brocard's problem is a problem in mathematics that seeks integer values of
More formally, it seeks pairs of integers
The problem was posed by Henri Brocard in a pair of articles in 1876 and 1885,[1][2] and independently in 1913 by Srinivasa Ramanujan.
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.
[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.
[5] Computational searches up to one quadrillion have found no further solutions.
[6][7][8] It would follow from the abc conjecture that there are only finitely many Brown numbers.
[9] More generally, it would also follow from the abc conjecture that
has only finitely many solutions, for any given integer
has only finitely many integer solutions, for any given polynomial
of degree at least 2 with integer coefficients.