 

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. 


Abstract
Given 3n algebraic integers α_{i,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
Σ_{i=1}^{n}Π_{ν}^{2}α_{i,ν}^{xν}≡ 0 (mod q), 1 ≤ x ≤ N. We then apply these bounds to studying arithmetic properties of values of
linear recurrence sequences on squares. 

Acknowledgements
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
igor.shparlinski@unsw.edu.au
Umberto Zannier:
Scuola Normale Superiore
Piazza dei Cavalieri, 7
56126 Pisa, Italy
u.zannier@sns.it

