Exponential sums and congruences with factorials

Moubariz Z. Garaev*, Florian Luca, Igor E. Shparlinski

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

13 Citations (Scopus)


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! + n1! + ⋯ +n47! with max{m, n, n1,..., n47} <p1350/1351+ε.

Original languageEnglish
Pages (from-to)29-44
Number of pages16
JournalJournal fur die Reine und Angewandte Mathematik
Issue number584
Publication statusPublished - Jul 2005


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