Products of Small Integers in Residue Classes and Additive Properties of Fermat Quotients

Glyn Harman; Igor E. Shparlinski

doi:10.1093/imrn/rnv182

Products of Small Integers in Residue Classes and Additive Properties of Fermat Quotients

Glyn Harman, Igor E. Shparlinski^*

^*Corresponding author for this work

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

19 Citations (Scopus)

Abstract

We show that for any ϵ > 0 and a sufficiently large cube-free q, any reduced residue class modulo q can be represented as a product of 14 integers from the interval [1, q^{1/4 e1/2 + ϵ}]. The length of the interval is at the lower limit of what is possible before the Burgess bound on the smallest quadratic nonresidue is improved. We also consider several variations of this result and give applications to Fermat quotients.

Original language	English
Pages (from-to)	1424-1446
Number of pages	23
Journal	International Mathematics Research Notices
Volume	2016
Issue number	5
DOIs	https://doi.org/10.1093/imrn/rnv182
Publication status	Published - 19 Jun 2016
Externally published	Yes

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10.1093/imrn/rnv182

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AB - We show that for any ϵ > 0 and a sufficiently large cube-free q, any reduced residue class modulo q can be represented as a product of 14 integers from the interval [1, q1/4 e1/2 + ϵ]. The length of the interval is at the lower limit of what is possible before the Burgess bound on the smallest quadratic nonresidue is improved. We also consider several variations of this result and give applications to Fermat quotients.

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Products of Small Integers in Residue Classes and Additive Properties of Fermat Quotients

Abstract

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