New York Journal of Mathematics
Volume 25 (2019), 207-218


Igor E. Shparlinski and Umberto Zannier

Arithmetic properties of quadratic exponential polynomials

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Published: February 15, 2019.
Keywords: exponential polynomials, congruences, linear recurrence sequences.
Subject: 11B37, 11D61.

Given 3n algebraic integers αi,ν, i=1...n, ν=0,1,2, and an integer ideal q in an algebraic number field K, we obtain several new bounds on the number of solutions to the congruence with a quadratic exponential polynomial
Σi=1nΠν2αi,νxν≡ 0 (mod q), 1 ≤ x ≤ N. We then apply these bounds to studying arithmetic properties of values of linear recurrence sequences on squares.


I.S. was supported in part by the ARC Grant DP180100201.

Author information

Igor E. Shparlinski:
Department of Pure Mathematics
University of New South Wales
Sydney, NSW 2052, Australia


Umberto Zannier:
Scuola Normale Superiore
Piazza dei Cavalieri, 7
56126 Pisa, Italy