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].

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 On the least square-free primitive root modulo p

2017 ◽  
Vol 170 ◽  
pp. 10-16 ◽  
Author(s):  
Stephen D. Cohen ◽  
Tim Trudgian
Keyword(s):  
Primitive Root ◽  
Least Square ◽  
Primitive Root Modulo ◽  
Modulo P

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 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.

On the Average of the Least Primitive Root Modulo p

1997 ◽  
Vol 56 (3) ◽  
pp. 435-454 ◽  
Author(s):  
P. D. T. A. Elliott ◽  
Leo Murata
Keyword(s):  
Primitive Root ◽  
Primitive Root Modulo ◽  
Modulo P

An Efficient Algorithm for Computing the k-Error Linear Complexity Spectrum of Periodic Sequences

2012 ◽  
Vol 532-533 ◽  
pp. 1726-1731
Author(s):  
Ling Yong Ma ◽  
Hao Cao
Keyword(s):  
Efficient Algorithm ◽  
Linear Complexity ◽  
Primitive Root ◽  
Periodic Sequences ◽  
Period 2 ◽  
Primitive Root Modulo ◽  
Modulo P ◽  
Error Linear Complexity ◽  
Error Linear Complexity Spectrum

An efficient algorithm for computing the k-error linear complexity spectrum of a q- ary sequence s with period 2 pn is presented, where q is an odd prime and a primitive root modulo p2. The algorithm generalizes both the Wei-Xiao-Chen and the Wei algorithms, The new algorithm can compute the k-error linear complexity spectrum of s using at most 4 n+1 steps.

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 Character sums for primitive root densities

2014 ◽  
Vol 157 (3) ◽  
pp. 489-511 ◽  
Author(s):  
H. W. LENSTRA ◽  
P. STEVENHAGEN ◽  
P. MOREE
Keyword(s):  
Galois Representations ◽  
Multiplicative Group ◽  
Primitive Root ◽  
Infinite Product ◽  
Character Sums ◽  
Correction Factors ◽  
Cyclic Reduction ◽  
Local Factors ◽  
Artin's Conjecture ◽  
Primitive Root Modulo

AbstractIt follows from the work of Artin and Hooley that, under assumption of the generalised Riemann hypothesis, the density of the set of primes q for which a given non-zero rational number r is a primitive root modulo q can be written as an infinite product ∏p δp of local factors δp reflecting the degree of the splitting field of Xp - r at the primes p, multiplied by a somewhat complicated factor that corrects for the ‘entanglement’ of these splitting fields.We show how the correction factors arising in Artin's original primitive root problem and several of its generalisations can be interpreted as character sums describing the nature of the entanglement. The resulting description in terms of local contributions is so transparent that it greatly facilitates explicit computations, and naturally leads to non-vanishing criteria for the correction factors.The method not only applies in the setting of Galois representations of the multiplicative group underlying Artin's conjecture, but also in the GL2-setting arising for elliptic curves. As an application, we compute the density of the set of primes of cyclic reduction for Serre curves.

scholarly journals The least primitive root modulo p2

2020 ◽  
Vol 215 ◽  
pp. 20-27
Author(s):  
Bryce Kerr ◽  
Kevin J. McGown ◽  
Tim Trudgian
Keyword(s):  
Primitive Root ◽  
Primitive Root Modulo

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.

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