Factorial-like values in the balancing sequence

Sa？etak 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