Existence of the solutions to the Brocard–Ramanujan problem for norm forms
Wataru Takeda
Source 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 .
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