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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Cuntz, Joachim / Huybrechts, Daniel / Hwang, Jun-Muk

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Exponential sums and congruences with factorials

Moubariz Z. Garaev1 / Florian Luca2 / Igor E. Shparlinski3




Citation Information: Journal für die reine und angewandte Mathematik. Volume 2005, Issue 584, Pages 29–44, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/crll.2005.2005.584.29, November 2005

Publication History

12. März 2004
Published Online:


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{mnn 1, … , n 47} < p 1350/1351+ ε .

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