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1
Content available remote

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

100%
Shparlinski I.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2008
|
tom 56
|
nr 1
1-7
EN
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 $𝓗_{a,p}(X,Y) = {(x,y) : xy ≡ a(mod p), 1 ≤x≤X, 1 ≤y≤Y}$. We give asymptotic formulas for the average values $∑_{\substack (x,y)∈ 𝓗_{a,p}(X,Y) x ≠ y*} φ(|x-y|)/|x-y|$ and $∑_{\substack (x,y)∈ 𝓗_{a,p}(X,X) x ≠ y*} φ(|x-y|)$ with the Euler function φ(k) on the differences between the components of points of $𝓗_{a,p}(X,Y)$.
2
Content available remote

Exponential Sums with Farey Fractions

100%
Shparlinski I.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2009
|
tom 57
|
nr 2
101-107
EN
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).
3
Content available remote

Primitive Points on a Modular Hyperbola

100%
Shparlinski I.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2006
|
tom 54
|
nr 3
193-200
EN
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.
4
Content available remote

On the concentration of points on modular hyperbolas and exponential curves

64%
Chan T., Shparlinski I.
Acta Arithmetica
|
2010
|
tom 142
|
nr 1
59-66
5
Content available remote

Pseudoprime Cullen and Woodall numbers

64%
Luca F., Shparlinski I.
Colloquium Mathematicum
|
2007
|
tom 107
|
nr 1
35-43
EN
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.
6
Content available remote

On Fully Split Lacunary Polynomials in Finite Fields

64%
Bibak K., Shparlinski I.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2011
|
tom 59
|
nr 3
197-202
EN
We estimate the number of possible degree patterns of k-lacunary polynomials of degree t < p which split completely modulo p. The result is based on a combination of a bound on the number of zeros of lacunary polynomials with some graph theory arguments.
7
Content available remote

Visible Points on Modular Exponential Curves

64%
Chan T., Shparlinski I.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2010
|
tom 58
|
nr 1
17-22
EN
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 $y ≡ ag^{x} (mod p)$. Among other tools, some recent results of additive combinatorics due to J. Bourgain and M. Z. Garaev play a crucial role in our argument.
8
Content available remote

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

64%
Sha M., Shparlinski I.
Acta Arithmetica
|
2015
|
tom 170
|
nr 4
299-325
EN
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. Some of the families we consider are much thinner than the ones previously studied.
9
Content available remote

Visible Points on Curves over Finite Fields

64%
Shparlinski I., Voloch J.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2007
|
tom 55
|
nr 3
193-199
EN
For a prime p and an absolutely irreducible modulo p polynomial f(U,V) ∈ ℤ[U,V] we obtain an asymptotic formula for the number of solutions to the congruence f(x,y) ≡ a (mod p) in positive integers x ≤ X, y ≤ Y, with the additional condition gcd(x,y) = 1. Such solutions have a natural interpretation as solutions which are visible from the origin. These formulas are derived on average over a for a fixed prime p, and also on average over p for a fixed integer a.
10
Content available remote

Prime numbers with Beatty sequences

64%
Banks W., Shparlinski I.
Colloquium Mathematicum
|
2009
|
tom 115
|
nr 2
147-157
EN
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 formula for the number of primes p = q⌊αn + β⌋ + a with n ≤ N, where α,β are real numbers such that α is positive and irrational of finite type (which is true for almost all α) and a,q are integers with $0 ≤ a < q ≤ N^κ$ and gcd(a,q) = 1, where κ > 0 depends only on α. We also prove a similar result for primes p = ⌊αn + β⌋ such that p ≡ a(mod q).
11
Content available remote

Twisted exponential sums over points of elliptic curves

64%
Ostafe A., Shparlinski I.
Acta Arithmetica
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2011
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tom 148
|
nr 1
77-92
12
Content available remote

On a ternary quadratic form over primes

64%
Fouvry É., Shparlinski I.
Acta Arithmetica
|
2011
|
tom 150
|
nr 3
285-314
13
Content available remote

Average Value of the Euler Function on Binary Palindromes

64%
Banks W., Shparlinski I.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2006
|
tom 54
|
nr 2
95-101
EN
We study values of the Euler function φ(n) taken on binary palindromes of even length. In particular, if $ℬ_{2ℓ}$ denotes the set of binary palindromes with precisely 2ℓ binary digits, we derive an asymptotic formula for the average value of the Euler function on $ℬ_{2ℓ}$.
14
Content available remote

On the distribution of the Euler function of shifted smooth numbers

64%
Loiperdinger S., Shparlinski I.
Colloquium Mathematicum
|
2010
|
tom 120
|
nr 1
139-148
EN
We give asymptotic formulas for some average values of the Euler function on shifted smooth numbers. The result is based on various estimates on the distribution of smooth numbers in arithmetic progressions which are due to A. Granville and É. Fouvry & G. Tenenbaum.
15
Content available remote

Arithmetic properties of numbers with restricted digits

64%
Banks W., Shparlinski I.
Acta Arithmetica
|
2004
|
tom 112
|
nr 4
313-332
16
Content available remote

Average multiplicative orders of elements modulo n

64%
Luca F., Shparlinski I.
Acta Arithmetica
|
2003
|
tom 109
|
nr 4
387-411
17
Content available remote

Distribution of consecutive modular roots of an integer

64%
Bourgain J., Shparlinski I.
Acta Arithmetica
|
2008
|
tom 134
|
nr 1
83-91
18
Content available remote

Exponential sums and the distribution of inversive congruential pseudorandom numbers with prime-power modulus

64%
Niederreiter H., Shparlinski I.
Acta Arithmetica
|
2000
|
tom 92
|
nr 1
89-98
19
Content available remote

On the smallest pseudopower

52%
Bourgain J., Konyagin S., Pomerance C., Shparlinski I.
Acta Arithmetica
|
2009
|
tom 140
|
nr 1
43-55
20
Content available remote

Some doubly exponential sums over ℤₘ

52%
Friedlander J., Konyagin S., Shparlinski I.
Acta Arithmetica
|
2002
|
tom 105
|
nr 4
349-370
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