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- W1738467033 abstract "In the present paper we obtain new upper bound estimates for the number of solutions of the congruence $$ xequiv y rpmod p;quad x,yin mathbb{N},quad x,yle H,quad rincU, $$ for certain ranges of $H$ and $|cU|$, where $cU$ is a subset of the field of residue classes modulo $p$ having small multiplicative doubling. We then use this estimate to show that the number of solutions of the congruence $$ x^nequiv lambdapmod p; quad xin N, quad L<x<L+p/n, $$ is at most $p^{frac{1}{3}-c}$ uniformly over positive integers $n, lambda$ and $L$, for some absolute constant $c>0$. This implies, in particular, that if $f(x)in Z[x]$ is a fixed polynomial without multiple roots in $C$, then the congruence $ x^{f(x)}equiv 1pmod p, ,xin mathbb{N}, ,xle p,$ has at most $p^{frac{1}{3}-c}$ solutions as $ptoinfty$, improving some recent results of Kurlberg, Luca and Shparlinski and of Balog, Broughan and Shparlinski. We use our results to show that almost all the residue classes modulo $p$ can be represented in the form $xg^y pmod p$ with positive integers $x<p^{5/8+varepsilon}$ and $y<p^{3/8}$. Here $g$ denotes a primitive root modulo $p$. We also prove that almost all the residue classes modulo $p$ can be represented in the form $xyzg^t pmod p$ with positive integers $x,y,z,t<p^{1/4+varepsilon}$." @default.
- W1738467033 created "2016-06-24" @default.
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- W1738467033 date "2016-01-14" @default.
- W1738467033 modified "2023-09-27" @default.
- W1738467033 title "Congruences involving product of intervals and sets with small multiplicative doubling modulo a prime and applications" @default.
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- W1738467033 doi "https://doi.org/10.1017/s0305004115000808" @default.
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