## Abstract

We estimate the number of solutions of certain diagonal congruences involving factorials. We use these results to bound exponential sums with products of two factorials *n*!*m*! and also derive asymptotic formulas for the number of solutions of various congruences with factorials. For example, we prove that the products of two factorials *n* !*m* ! with max{*n*,*m* } < *p*
^{1/2+}
* ^{ε}* are uniformly distributed modulo

*p*, and that any residue class modulo

*p*is representable in the form

*m*!

*n*! +

*n*

_{1}! + …+

*n*

_{47}! with max{

*m*,

*n*,

*n*

_{1}, … ,

*n*

_{47}} <

*p*

^{1350/1351+}

*.*

^{ε}
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