ON THE RANGES OF DISCRETE EXPONENTIALS

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 t at (mod p).


Introduction.
Let φ(X) be a formula in one free variable X, written in the firstorder language of rings.Then for every ring R with identity, φ(X) defines a subset of R consisting of all elements of R satisfying φ(X).For example, the formula (∃Y )(X = Y 2 ) will define in every ring R the set of perfect squares in R (for an introduction to the basic concepts arising in model theory of first-order languages, we refer to [5]).
The value sets (ranges) of polynomials over finite fields have been studied by various authors, and many interesting results have been proved (see [3, pages 379-381]).Note that if f (X) is a polynomial with integer coefficients, the formula (∃Y )(X = f (Y )) will define in every finite field F q the value set of the function from F q to F q induced by f .The value sets of the discrete exponentials are no less interesting.For example, if a > 1 is an integer that is not a square, Artin's conjecture for primitive roots [4] implies that the range of the function t → a t (modp) has p − 1 elements for infinitely many primes p.In the present note, we investigate the ranges of exponential functions from the point of view of definability.Note that the range of exp a : Z → F p coincides with a , the cyclic subgroup of F * p generated by a (modulo p).Our main result will be the following.
Theorem 1.1.Let a > 1 be a fixed integer.Then there is no formula φ(X) in one free variable X, written in the first-order language of rings, such that for any prime p with gcd(a, p) = 1, the set of all elements in the finite prime field F p satisfying φ coincides with the range of the discrete exponential exp a : Z → F p .
Here is a brief outline of the proof.We will first prove a result (Theorem 2.1) concerning the existence of primes with respect to which a fixed integer a > 1 has sufficiently small orders.This, in conjunction with a seminal result of Chatzidakis et al. [1] on definable subsets over finite fields, will lead to the proof of Theorem 1.1.

Small orders modulo p.
In what follows, we will prove that there exist infinitely many primes with respect to which a given integer a > 1 has "small order."More precisely, the following result holds true.
Theorem 2.1.Let a > 1 be an integer.Then, for every ε > 0, there exist infinitely many primes q such that ord q (a), the order of a modulo q, satisfies ord q (a) < qε. (2.1) Proof.Let k be an integer satisfying and let p be a prime satisfying Due to Dirichlet's theorem on primes in arithmetic progressions [2], there are infinitely many primes p satisfying (2.3) and (2.4).We select a prime q with the property (2.5) Note that both p and q are necessarily odd.Since from (2.5) it follows that a p ≡ 1(mod q), (2.6) the order ord q (a) can be either 1 or p.We will rule out the possibility ord q (a) = 1.Indeed, if ord q (a) = 1, then On the other hand, 1 with integer coefficients, and therefore From (2.5), (2.7), and (2.8) it follows q | p and, since p, q are primes, q = p.This, together with (2.7), leads us to p | a−1, and therefore a > p, which contradicts assumption (2.3).This leaves us with ord q (a) = p.
(2.9) From (2.9) and from a q−1 ≡ 1(mod q) it follows that p | q − 1, so that for some positive integer t.We will show that t > k, so that q > kp + 1.
(2.11) Indeed, we assume, for contradiction, that t ≤ k.From (2.4), we get p = (k + 1)!s + 1 for some positive integer s.Then (2.12) Note that t + 1 is, under the assumption t ≤ k, a divisor of (k + 1)!.Then, from (2.12), q will be a multiple of t + 1, a contradiction, since 2 ≤ t + 1 < q.Thus, (2.11) holds true and, consequently, since 1/k < ε, we get where the infimum is taken over all primes q > a.This completes the proof of Theorem 2.1.

Proof of the main result.
We now proceed to the proof of Theorem 1.1.We will use the following result which is a corollary of the main theorem in [ We are now ready to proceed to the proof of Theorem 1.1.Assume, for contradiction, that for some integer a > 1 there exists a first-order formula φ(X) in the language of rings such that for every prime p a, we have Remark 3.2.From Theorem 1.1, it follows as an immediate corollary that, if a > 1 is a fixed integer, then there is no first-order formula φ(X) in the first-order language of rings, such that for any prime p, the set of all elements in F p satisfying φ is {a t mod p | t ≥ 1}.Indeed, assuming such a formula exists, it would define in any F p with gcd(a, p) = 1 the range of the discrete exponential exp a : Z → F p .

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

. 1 )
From (3.1) we get φ F p = ord p (a) (3.2) for all p a. Clearly, ord p (a) > log a (p) (3.3) for all p a. From (3.2), (3.3), and Theorem 3.1, it follows that for every large enough prime p, we have ord p (a) ≥ Cp. (3.4)Clearly,(3.4)  is in contradiction to Theorem 2.1 proved above, which implies that