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International Journal of Mathematics and Mathematical Sciences
Volume 2004, Issue 42, Pages 2265-2268

On the ranges of discrete exponentials

1Department of Mathematics II, University Politechnica of Bucharest, Splaiul Independentei 313, Bucharest 77206, Romania
2Department of Mathematics, Ohio Northern University, Ada 45810, OH, USA

Received 3 December 2003

Copyright © 2004 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let a>1 be a fixed integer. We prove that there is no first-order formula ϕ(X) in one free variable X, written in the language of rings, such that for any prime p with gcd(a,p)=1 the set of all elements in the finite prime field Fp satisfying ϕ coincides with the range of the discrete exponential function tat(modp).