Exponential sums and congruences with factorials

Moubariz Z. Garaev; Florian Luca; Igor E. Shparlinski

Exponential sums and congruences with factorials

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

^*Corresponding author for this work

School of Computing

Research output: Contribution to journal › Article › peer-review

14 Citations (Scopus)

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

Original language	English
Pages (from-to)	29-44
Number of pages	16
Journal	Journal fur die Reine und Angewandte Mathematik
Issue number	584
Publication status	Published - Jul 2005

Cite this

@article{f3d25c04e00b4bd6b1f5fa0c43464e56,

title = "Exponential sums and congruences with factorials",

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

author = "Garaev, {Moubariz Z.} and Florian Luca and Shparlinski, {Igor E.}",

year = "2005",

month = jul,

language = "English",

pages = "29--44",

journal = "Journal fur die Reine und Angewandte Mathematik",

issn = "0075-4102",

publisher = "De Gruyter",

number = "584",

}

TY - JOUR

T1 - Exponential sums and congruences with factorials

AU - Garaev, Moubariz Z.

AU - Luca, Florian

AU - Shparlinski, Igor E.

PY - 2005/7

Y1 - 2005/7

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=27844551345&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:27844551345

SN - 0075-4102

SP - 29

EP - 44

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

IS - 584

ER -

Exponential sums and congruences with factorials

Abstract

Other files and links

Fingerprint

Cite this