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On the number of zero trace elements in polynomial bases for F.

Igor E. Shparlinski — 2005

Revista Matemática Complutense

Let F denote the finite field of q elements. O. Ahmadi and A. Menezes have recently considered the question about the possible number of elements with zero trace in polynomial bases of F over F. Here we show that the Weil bound implies that there is such a basis with n + O(log n) zero-trace elements.

On the Euler Function on Differences Between the Coordinates of Points on Modular Hyperbolas

Igor E. Shparlinski — 2008

Bulletin of the Polish Academy of Sciences. Mathematics

For a prime p > 2, an integer a with gcd(a,p) = 1 and real 1 ≤ X,Y < p, we consider the set of points on the modular hyperbola a , p ( X , Y ) = ( x , y ) : x y a ( m o d p ) , 1 x X , 1 y Y . We give asymptotic formulas for the average values ( x , y ) a , p ( X , Y ) x y * φ ( | x - y | ) / | x - y | and ( x , y ) a , p ( X , X ) x y * φ ( | x - y | ) with the Euler function φ(k) on the differences between the components of points of a , p ( X , Y ) .

Primitive Points on a Modular Hyperbola

Igor E. Shparlinski — 2006

Bulletin of the Polish Academy of Sciences. Mathematics

For positive integers m, U and V, we obtain an asymptotic formula for the number of integer points (u,v) ∈ [1,U] × [1,V] which belong to the modular hyperbola uv ≡ 1 (mod m) and also have gcd(u,v) =1, which are also known as primitive points. Such points have a nice geometric interpretation as points on the modular hyperbola which are "visible" from the origin.

Exponential Sums with Farey Fractions

Igor E. Shparlinski — 2009

Bulletin of the Polish Academy of Sciences. Mathematics

For positive integers m and N, we estimate the rational exponential sums with denominator m over the reductions modulo m of elements of the set ℱ(N) = {s/r : r,s ∈ ℤ, gcd(r,s) = 1, N ≥ r > s ≥ 1} of Farey fractions of order N (only fractions s/r with gcd(r,m) = 1 are considered).

Small discriminants of complex multiplication fields of elliptic curves over finite fields

Igor E. Shparlinski — 2015

Czechoslovak Mathematical Journal

We obtain a conditional, under the Generalized Riemann Hypothesis, lower bound on the number of distinct elliptic curves E over a prime finite field 𝔽 p of p elements, such that the discriminant D ( E ) of the quadratic number field containing the endomorphism ring of E over 𝔽 p is small. For almost all primes we also obtain a similar unconditional bound. These lower bounds complement an upper bound of F. Luca and I. E. Shparlinski (2007).

On the Győry-Sárközy-Stewart conjecture in function fields

Igor E. Shparlinski — 2018

Czechoslovak Mathematical Journal

We consider function field analogues of the conjecture of Győry, Sárközy and Stewart (1996) on the greatest prime divisor of the product ( a b + 1 ) ( a c + 1 ) ( b c + 1 ) for distinct positive integers a , b and c . In particular, we show that, under some natural conditions on rational functions F , G , H ( X ) , the number of distinct zeros and poles of the shifted products F H + 1 and G H + 1 grows linearly with deg H if deg H max { deg F , deg G } . We also obtain a version of this result for rational functions over a finite field.

On the largest prime factor of n ! + 2 n - 1

Florian LucaIgor E. Shparlinski — 2005

Journal de Théorie des Nombres de Bordeaux

For an integer n 2 we denote by P ( n ) the largest prime factor of n . We obtain several upper bounds on the number of solutions of congruences of the form n ! + 2 n - 1 0 ( mod q ) and use these bounds to show that lim sup n P ( n ! + 2 n - 1 ) / n ( 2 π 2 + 3 ) / 18 .

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