scholarly journals On Minimal Sets of Generators for Primitive Roots

1995 ◽  
Vol 38 (4) ◽  
pp. 465-468 ◽  
Author(s):  
Francesco Pappalardi
Keyword(s):  
Primitive Root ◽  
Minimal Sets ◽  
Almost All ◽  
Primitive Roots

AbstractA conjecture of Brown and Zassenhaus (see [2]) states that the first log/? primes generate a primitive root (mod p) for almost all primes p. As a consequence of a Theorem of Burgess and Elliott (see [3]) it is easy to see that the first log2p log log4+∊p primes generate a primitive root (mod p) for almost all primes p. We improve this showing that the first log2p/ log log p primes generate a primitive root (mod p) for almost all primes p.

scholarly journals The Jones polynomial: quantum algorithms and applications in quantum complexity theory

2008 ◽  
Vol 8 (1&2) ◽  
pp. 147-180
Author(s):  
P. Wocjan ◽  
J. Yard
Keyword(s):  
Quantum Computation ◽  
Braid Group ◽  
Quantum Algorithms ◽  
Primitive Root ◽  
Jones Polynomial ◽  
Tutte Polynomial ◽  
Roots Of Unity ◽  
Complete Problems ◽  
Primitive Roots ◽  
Quantum Complexity

We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for approximately evaluating a family of specializations of the HOMFLYPT two-variable polynomial of trace closures of braids. We also give algorithms for approximating the Jones polynomial of a general class of closures of braids at roots of unity. Next we provide a self-contained proof of a result of Freedman et al.\ that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. Our proof encodes two-qubit unitaries into the rectangular representation of the eight-strand braid group. We then give QCMA-complete and PSPACE-complete problems which are based on braids. We conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a \#P-hard problem, while learning its most significant bit is PP-hard, circumventing the usual route through the Tutte polynomial and graph coloring.

Explicit upper bound on the least primitive root modulo p2

2021 ◽  
pp. 1-7
Author(s):  
Bo Chen
Keyword(s):  
Recent Result ◽  
Upper Bound ◽  
Primitive Root ◽  
Character Sums ◽  
Primitive Root Modulo ◽  
Primitive Roots

In this paper, we give an explicit upper bound on [Formula: see text], the least primitive root modulo [Formula: see text]. Since a primitive root modulo [Formula: see text] is not primitive modulo [Formula: see text] if and only if it belongs to the set of integers less than [Formula: see text] which are [Formula: see text]th power residues modulo [Formula: see text], we seek the bounds for [Formula: see text] and [Formula: see text] to find [Formula: see text] which satisfies [Formula: see text], where, [Formula: see text] denotes the number of primitive roots modulo [Formula: see text] not exceeding [Formula: see text], and [Formula: see text] denotes the number of [Formula: see text]th powers modulo [Formula: see text] not exceeding [Formula: see text]. The method we mainly use is to estimate the character sums contained in the expressions of the [Formula: see text] and [Formula: see text] above. Finally, we show that [Formula: see text] for all primes [Formula: see text]. This improves the recent result of Kerr et al.

scholarly journals Optimasi Kinerja Primitive Root Diffuser (PRD) dengan Teknik Sisipan Resonator Jamak (Halaman 25 s.d. 29)

2015 ◽  
Vol 18 (52) ◽  
Author(s):  
Fahrudin Ahmad ◽  
Harjana ◽  
Suparmi ◽  
Iwan Yahya
Keyword(s):  
Primitive Root ◽  
Primitive Roots

Telah  dilakukan kajian untuk meningkatkan kinerja primitive roots diffuser (PRD) dengan model standar berbentuk Skyline dengan jumlah elemen 5x8. Teknik modifikasi yang telah diterapkan adalah penyisipan rangkaian resonator seperempat panjang gelombang dengan kedalaman yang bersesuaian dengan masing-masing elemen PRD. Pengujian dilaksanakan dengan metode interrupted noise mengacu kepada prosedur standar ISO 3382 untuk mendapatkan nilai waktu dengung T30 dan T20. Adapun respon spasial dari PRD yang dikembangkan disimulasikan dengan perangkat lunak AFMG Reflex. Hasil pengujian menunjukkan bahwa penambahan rangkaian resonator jamak berhasil mengurangi nilai waktu dengung rerata sebesar 0,78 dan 1,83 sekon berturut-turut untuk T30 dan T20 pada bentang frekuensi rendah. Di samping itu pemakaian resonator memperbaiki respon spasial omnidireksional PRD. 

scholarly journals Congruences involving product of intervals and sets with small multiplicative doubling modulo a prime and applications

2016 ◽  
Vol 160 (3) ◽  
pp. 477-494 ◽  
Author(s):  
J. CILLERUELO ◽  
M. Z. GARAEV
Keyword(s):  
Upper Bound ◽  
Absolute Constant ◽  
Primitive Root ◽  
Multiple Roots ◽  
Number Of Solutions ◽  
Residue Classes ◽  
Primitive Root Modulo ◽  
Image Position ◽  
Almost All ◽  
Positive Integers

AbstractIn this paper we obtain new upper bound estimates for the number of solutions of the congruence $$\begin{equation} x\equiv y r\pmod p;\quad x,y\in \mathbb{N},\quad x,y\le H,\quad r\in \mathcal{U}, \end{equation}$$ for certain ranges of H and |${\mathcal U}$|, where ${\mathcal U}$ is a subset of the field of residue classes modulo p having small multiplicative doubling. We then use these estimates to show that the number of solutions of the congruence $$\begin{equation} x^n\equiv \lambda\pmod p; \quad x\in \mathbb{N}, \quad L<x<L+p/n, \end{equation}$$ is at most $p^{\frac{1}{3}-c}$ uniformly over positive integers n, λ and L, for some absolute constant c > 0. This implies, in particular, that if f(x) ∈ $\mathbb{Z}$[x] is a fixed polynomial without multiple roots in $\mathbb{C}$, then the congruence xf(x) ≡ 1 (mod p), x ∈ $\mathbb{N}$, x ⩽ p, has at most $p^{\frac{1}{3}-c}$ solutions as p → ∞, improving some recent results of Kurlberg, Luca and Shparlinski and of Balog, Broughan and Shparlinski. We use our results to show that almost all the residue classes modulo p can be represented in the form xgy (mod p) with positive integers x < p5/8+ϵ and y < p3/8. Here g denotes a primitive root modulo p. We also prove that almost all the residue classes modulo p can be represented in the form xyzgt (mod p) with positive integers x, y, z, t < p1/4+ϵ.

scholarly journals The settling-time reducibility ordering

2007 ◽  
Vol 72 (3) ◽  
pp. 1055-1071 ◽  
Author(s):  
Barbara F. Csima ◽  
Richard A. Shore
Keyword(s):  
Differential Geometry ◽  
Natural Extension ◽  
Computable Function ◽  
Settling Time ◽  
Partial Ordering ◽  
Minimal Sets ◽  
Last Stage ◽  
Almost All ◽  
Computable Isomorphism

AbstractTo each computable enumerable (c.e.) set A with a particular enumeration {As}s∈ω there is associated a settling function mA(x), where mA(x) is the last stage when a number less than or equal to x was enumerated into A. One c.e. set A is settling time dominated by another set B (B >stA) if for every computable function f, for all but finitely many x, mB(x) > f(mA(x)). This settling-time ordering, which is a natural extension to an ordering of the idea of domination, was first introduced by Nabutovsky and Weinberger in [3] and Soare [6]. They desired a sequence of sets descending in this relationship to give results in differential geometry. In this paper we examine properties of the <st ordering. We show that it is not invariant under computable isomorphism, that any countable partial ordering embeds into it. that there are maximal and minimal sets, and that two c.e. sets need not have an inf or sup in the ordering. We also examine a related ordering, the strong settling-time ordering where we require for all computable f and g, for almost all x, mB(x) > f(mA(g(x))).

scholarly journals IV. On the residues of powers of numbers any composite modulus, real or complex

1893 ◽  
Vol 184 ◽  
pp. 189-336
Keyword(s):  
Real Number ◽  
Primitive Root ◽  
Real Numbers ◽  
Composite Modulus ◽  
Composite Number ◽  
Complete Set ◽  
Primitive Roots

The present work consists of two. parts, with an appendix to the second. Part I. deals with real numbers, Part II. with complex. In the simple cases when the modulus is a real number, which is an odd prime, a power of an odd prime, or double the power of an odd prime, we know that there exist primitive roots of the modulus ; that is, that there are numbers whose successive powers have for their residues the complete set of numbers less than and prime to the modulus. A primitive root may be said to generate by its successive powers the complete set of residues. It is also known that, in general, when the modulus is any composite number, though primitive roots do not exist, there may be laid down a set of numbers which will here be called g, the products of powers of which give the complete set of residues prime to the modulus.

scholarly journals The Generalized Artin Primitive Root Conjecture

2018 ◽  
Vol 11 (1) ◽  
pp. 23
Author(s):  
N. A. Carella
Keyword(s):  
Primitive Root ◽  
Asymptotic Formulas ◽  
Artin Primitive Root Conjecture ◽  
Primitive Roots

Asymptotic formulas for the number of integers with the primitive root 2, and the generalized Artin conjecture for subsets of composite integers with fixed admissible primitive roots \(u\neq \pm 1,v^2\), are presented here.

scholarly journals Note on Artin's Conjecture on Primitive Roots

2022 ◽  
Vol Volume 44 - Special... ◽  
Author(s):  
Sankar Sitaraman
Keyword(s):  
Primitive Root ◽  
Artin's Conjecture ◽  
Multiplicative Order ◽  
Primitive Root Modulo ◽  
Modulo P ◽  
Primitive Roots

E. Artin conjectured that any integer $a > 1$ which is not a perfect square is a primitive root modulo $p$ for infinitely many primes $ p.$ Let $f_a(p)$ be the multiplicative order of the non-square integer $a$ modulo the prime $p.$ M. R. Murty and S. Srinivasan \cite{Murty-Srinivasan} showed that if $\displaystyle \sum_{p < x} \frac 1 {f_a(p)} = O(x^{1/4})$ then Artin's conjecture is true for $a.$ We relate the Murty-Srinivasan condition to sums involving the cyclotomic periods from the subfields of $\mathbb Q(e^{2\pi i /p})$ corresponding to the subgroups $<a> \subseteq \mathbb F_p^*.$

scholarly journals Square-full primitive roots

2018 ◽  
Vol 14 (04) ◽  
pp. 1013-1021 ◽  
Author(s):  
Marc Munsch ◽  
Tim Trudgian
Keyword(s):  
Primitive Root ◽  
Least Square ◽  
Character Sum ◽  
Primitive Root Modulo ◽  
Primitive Roots

We use character sum estimates to give some bounds on the least square-full primitive root modulo a prime. In particular, we show that there is a square-full primitive root mod [Formula: see text] less than [Formula: see text].

scholarly journals PRIMITIVE ROOTS IN QUADRATIC FIELDS

2006 ◽  
Vol 02 (01) ◽  
pp. 7-23 ◽  
Author(s):  
JOSEPH COHEN
Keyword(s):  
Riemann Hypothesis ◽  
Primitive Root ◽  
Maximal Order ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Generalized Riemann Hypothesis ◽  
Artin's Conjecture ◽  
Modulo P ◽  
Primitive Roots ◽  
Real Quadratic

We consider an analogue of Artin's primitive root conjecture for units in real quadratic fields. Given such a nontrivial unit, for a rational prime p which is inert in the field the maximal order of the unit modulo p is p + 1. An extension of Artin's conjecture is that there are infinitely many such inert primes for which this order is maximal. This is known at present only under the Generalized Riemann Hypothesis. Unconditionally, we show that for any choice of 7 units in different real quadratic fields satisfying a certain simple restriction, there is at least one of the units which satisfies the above version of Artin's conjecture.

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