Products of three factorials (Q2875457)

In a classical paper, \textit{P. Erdős} and \textit{R. L. Graham} [Bull. Inst. Math., Acad. Sin. 4, 337--355 (1976; Zbl 0346.10004)] investigated square values of products of factorials, and posed several questions and problems. One of their questions reduces to deciding whether the product \(a_1!(a_1-3)!a_3!\) allows other square values than \(10!7!6!\), \(50!47!3!\), \(50!47!4!\).NEWLINENEWLINE In the present paper the authors solve the equation NEWLINE\[NEWLINE a_1!(a_1-3)!a_3!=y^2 NEWLINE\]NEWLINE for all values of \(a_3\) with \(a_3\leq 100\). They obtain that in these cases all solutions are given by NEWLINE\[NEWLINE (a_1, a_3) \in \{(10, 6),(50, 3),(50, 4),(324, 26),(352, 13),(442, 18),(2738, 26)\}. NEWLINE\]NEWLINE Hence, in particular, they extend the set found by Erdős and Graham (loc. cit.). Beside this, they derive some properties for the solutions of the above equation in the general case, where \(a_3\) is arbitrary.