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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 .

Prime numbers with Beatty sequences

William D. BanksIgor E. Shparlinski — 2009

Colloquium Mathematicae

A study of certain Hamiltonian systems has led Y. Long to conjecture the existence of infinitely many primes which are not of the form p = 2⌊αn⌋ + 1, where 1 < α < 2 is a fixed irrational number. An argument of P. Ribenboim coupled with classical results about the distribution of fractional parts of irrational multiples of primes in an arithmetic progression immediately implies that this conjecture holds in a much more precise asymptotic form. Motivated by this observation, we give an asymptotic...

Pseudoprime Cullen and Woodall numbers

Florian LucaIgor E. Shparlinski — 2007

Colloquium Mathematicae

We show that if a > 1 is any fixed integer, then for a sufficiently large x>1, the nth Cullen number Cₙ = n2ⁿ +1 is a base a pseudoprime only for at most O(x log log x/log x) positive integers n ≤ x. This complements a result of E. Heppner which asserts that Cₙ is prime for at most O(x/log x) of positive integers n ≤ x. We also prove a similar result concerning the pseudoprimality to base a of the Woodall numbers given by Wₙ = n2ⁿ - 1 for all n ≥ 1.

Visible Points on Modular Exponential Curves

Tsz Ho ChanIgor E. Shparlinski — 2010

Bulletin of the Polish Academy of Sciences. Mathematics

We obtain an asymptotic formula for the number of visible points (x,y), that is, with gcd(x,y) = 1, which lie in the box [1,U] × [1,V] and also belong to the exponential modular curves y a g x ( m o d p ) . Among other tools, some recent results of additive combinatorics due to J. Bourgain and M. Z. Garaev play a crucial role in our argument.

Lang-Trotter and Sato-Tate distributions in single and double parametric families of elliptic curves

Min ShaIgor E. Shparlinski — 2015

Acta Arithmetica

We obtain new results concerning the Lang-Trotter conjectures on Frobenius traces and Frobenius fields over single and double parametric families of elliptic curves. We also obtain similar results with respect to the Sato-Tate conjecture. In particular, we improve a result of A. C. Cojocaru and the second author (2008) towards the Lang-Trotter conjecture on average for polynomially parameterised families of elliptic curves when the parameter runs through a set of rational numbers of bounded height....

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