### Factorial-like values in the balancing sequence

#### Abstract

In this paper, we solve a few diophantine equations linked to balancing numbers and factorials. The basic problem $B_y=x!$ has only the one nontrivial solution $B_2=6=3!$, but it is a direct consequence of a theorem of F.~Luca. The more difficult problem $B_y=x_2!/x_1!$ is still open, but we solve it under different conditions. Two related problems are also studied.

#### Keywords

Factorials, balancing numbers, diophantine equation

#### Full Text:

PDFISSN: 1331-0623 (Print), 1848-8013 (Online)