Squares and factorials in products of factorials
The authors prove several results in connection with a problem of Erdős and Graham concerning products of factorials yielding a factorial or a perfect square. They considerably improve upon an estimate of Erdős and Graham concerning the number of those positive integers \(n\) up to \(X\), which are not squares, but \(n!a!b!\) is a square with some integers \(a,b\) with \(n\geq a\geq b\). On their way, they also improve upon a related estimate of Erdős and Graham concerning those integers \(n\) for which the exponent of the largest prime factor in the product of a block of consecutive integers starting with \(n\), is at least two. Finally, they derive upper bounds for the solutions of the equation \(a_2!\cdots a_t!=m(m+1)\cdots (m+k-1)\), both under the abc-conjecture and unconditionally. To prove their theorems, the authors combine several results of Erdős and Graham with new ideas and tools from prime number theory in a skillful and sophisticated way.
scientific article; zbMATH DE number 6369395