Existence of the solutions to the Brocard–Ramanujan problem for norm forms

Abstract

The Brocard–Ramanujan problem, which is an unsolved problem in number theory, is to find integer solutions ( x , ℓ ) (x,\ell ) of x 2 − 1 = ℓ ! x^2-1=\ell ! . Many analogs of this problem are currently being considered. As one example, it is known that there are at most only finitely many algebraic integer solutions ( x , ℓ ) (x, \ell ) , up to a unit factor, to the equations N K ( x ) = ℓ ! N_K(x) = \ell ! , where N K N_K are the norms of number fields K / Q K/\mathbf Q . In this paper, we construct infinitely many number fields K K such that N K ( x ) = ℓ ! N_K(x) = \ell ! has at least 22 22 solutions for positive integers ℓ \ell .