The Experts below are selected from a list of 888 Experts worldwide ranked by ideXlab platform

Dongxi Ye - One of the best experts on this subject based on the ideXlab platform.

Zhi-wei Sun - One of the best experts on this subject based on the ideXlab platform.

  • on some determinants with Legendre Symbol entries
    Finite Fields and Their Applications, 2019
    Co-Authors: Zhi-wei Sun
    Abstract:

    Abstract In this paper we mainly focus on some determinants with Legendre Symbol entries. Let p be an odd prime and let ( ⋅ p ) be the Legendre Symbol. We show that ( − S ( d , p ) p ) = 1 for any d ∈ Z with ( d p ) = 1 , and that ( W p p ) = { ( − 1 ) | { 0 k p 4 : ( k p ) = − 1 } | if p ≡ 1 ( mod 4 ) , ( − 1 ) ⌊ ( p + 1 ) / 8 ⌋ if p ≡ 3 ( mod 4 ) , where S ( d , p ) = det ⁡ [ ( i 2 + d j 2 p ) ] 1 ⩽ i , j ⩽ ( p − 1 ) / 2 and W p = det ⁡ [ ( i 2 − ( ( p − 1 ) / 2 ) ! j p ) ] 0 ⩽ i , j ⩽ ( p − 1 ) / 2 . We also pose some conjectures on determinants, one of which states that ( − 1 ) ⌊ ( p + 1 ) / 8 ⌋ W p is a square when p ≡ 3 ( mod 4 ) .

  • on some determinants with Legendre Symbol entries
    arXiv: Number Theory, 2013
    Co-Authors: Zhi-wei Sun
    Abstract:

    In this paper we mainly focus on some determinants with Legendre Symbol entries. Let $p$ be an odd prime and let $(\frac{\cdot}p)$ be the Legendre Symbol. We show that $(\frac{-S(d,p)}p)=1$ for any $d\in\mathbb Z$ with $(\frac dp)=1$, and that $$\left(\frac{W_p}p\right)=\begin{cases}(-1)^{|\{0determinants, one of which states that $(-1)^{\lfloor(p+1)/8\rfloor}W_p$ is a square when $p\equiv 3\pmod4$.

  • congruences concerning Legendre polynomials ii
    Journal of Number Theory, 2013
    Co-Authors: Zhi-wei Sun
    Abstract:

    Let p>3 be a prime, and let m be an integer with p∤m. In the paper we solve some conjectures of Z.W. Sun concerning ∑k=0p−1(2kk)3/mk(modp2), ∑k=0p−1(2kk)(4k2k)/mk(modp) and ∑k=0p−1(2kk)2(4k2k)/mk(modp2). In particular, we show that ∑k=0p−12(2kk)3≡0(modp2) for p≡3,5,6(mod7). Let {Pn(x)} be the Legendre polynomials. In the paper we also show that P[p4](t)≡−(6p)∑x=0p−1(x3−32(3t+5)x+9t+7p)(modp), where t is a rational p-adic integer, [x] is the greatest integer not exceeding x and (ap) is the Legendre Symbol. As consequences we determine P[p4](t)(modp) in the cases t=−53,−79,−6563 and confirm many conjectures of Z.W. Sun.

  • ON SOME DETERMINANTS WITH Legendre Symbol ENTRIES
    2013
    Co-Authors: Zhi-wei Sun
    Abstract:

    In this paper we mainly focus on some determinants with Legendre Symbol entries. For an odd prime p and an integer d, let S(d,p) denote the determinant of the (p − 1)/2 × (p − 1)/2 matrix whose (i,j)-entry (1 � i,j � (p−1)/2) is the Legendre Symbol ( i2 +dj 2). We investigate properties of S(d,p) p as well as some other determinants involving Legendre Symbols. In Section 3 we pose over ten open conjectures on determinants one of which states that ( −S(d,p)) = 1 if

  • on some new congruences for binomial coefficients
    International Journal of Number Theory, 2011
    Co-Authors: Zhi-wei Sun, Roberto Tauraso
    Abstract:

    In this paper we establish some new congruences involving central binomial coefficients as well as Catalan numbers. Let p be a prime and let a be any positive integer. We determine for d = 0, …, pa and for δ = 0, 1. We also show that for every n = 0, 1, 2, …, where Cm is the Catalan number , and is the Legendre Symbol.

Shaun Cooper - One of the best experts on this subject based on the ideXlab platform.

Sun Zhi-wei - One of the best experts on this subject based on the ideXlab platform.

  • A new theorem on quadratic residues modulo primes
    2021
    Co-Authors: Hou Qing-hu, Pan Hao, Sun Zhi-wei
    Abstract:

    Let $p>3$ be a prime, and let $(\frac{\cdot}p)$ be the Legendre Symbol. Let $b\in\mathbb Z$ and $\varepsilon\in\{\pm 1\}$. We mainly prove that $$\left|\left\{N_p(a,b):\ 1

  • Supercongruences for central trinomial coefficients
    2020
    Co-Authors: Pan Hao, Sun Zhi-wei
    Abstract:

    For each $n=0,1,2,\ldots$ the central trinomial coefficient $T_n$ is the coefficient of $x^n$ in the expansion of $(x^2+x+1)^n$. In 2016 the second author conjectured that for any prime $p>3$ and positive integer $n$ the quotient $(T_{pn}-T_n)/(pn)^2$ is a $p$-adic integer. In this paper we confirm this conjecture and prove further that $$\frac{T_{pn}-T_n}{(pn)^2}\equiv\frac{T_{n-1}}6\left(\frac p3\right)B_{p-2}\left(\frac13\right)\pmod p,$$ where $(\frac p3)$ is the Legendre Symbol and $B_{p-2}(x)$ is the Bernoulli polynomial of degree $p-2$.Comment: 9 page

  • New observations on primitive roots modulo primes
    2020
    Co-Authors: Sun Zhi-wei
    Abstract:

    We make many new observations on primitive roots modulo primes. For an odd prime $p$ and an integer $c$, we establish a theorem concerning $\sum_g(\frac{g+c}p)$, where $g$ runs over all the primitive roots modulo $p$ among $1,\ldots,p-1$, and $(\frac{\cdot}p)$ denotes the Legendre Symbol. On the basis of our numerical computations, we formulate 35 conjectures involving primitive roots modulo primes. For example, we conjecture that for any prime $p$ there is a primitive root $g3$ there is a prime $q3$ there exists a Fibonacci number $F_k3$.Comment: 23 page

  • Proof of three conjectures on determinants related to quadratic residues
    2020
    Co-Authors: Grinberg Darij, Sun Zhi-wei, Zhao Lilu
    Abstract:

    In this paper we confirm three conjectures of Z.-W. Sun on determinants. We first show that any odd integer $n>3$ divides the determinant $$\left|(i^2+dj^2)\left(\frac{i^2+dj^2}n\right)\right|_{0\le i,j\le (n-1)/2},$$ where $d$ is any integer and $(\frac{\cdot}n)$ is the Jacobi Symbol. Then we prove some divisibility results concerning $|(i+dj)^n|_{0\le i,j\le n-1}$ and $|(i^2+dj^2)^n|_{0\le i,j\le n-1}$, where $d\not=0$ and $n>2$ are integers. Finally, for any odd prime $p$ and integers $c$ and $d$ with $p\nmid cd$, we determine completely the Legendre Symbol $(\frac{S_c(d,p)}p)$, where $S_c(d,p):=|(\frac{i^2+dj^2+c}p)|_{1\le i,j\le(p-1)/2}$.Comment: 14 pages, accepted by Linear and Multilinear Algebr

  • On some determinants with Legendre Symbol entries
    2019
    Co-Authors: Sun Zhi-wei
    Abstract:

    In this paper we mainly focus on some determinants with Legendre Symbol entries. Let $p$ be an odd prime and let $(\frac{\cdot}p)$ be the Legendre Symbol. We show that $(\frac{-S(d,p)}p)=1$ for any $d\in\mathbb Z$ with $(\frac dp)=1$, and that $$\left(\frac{W_p}p\right)=\begin{cases}(-1)^{|\{0

Uehara Satoshi - One of the best experts on this subject based on the ideXlab platform.

  • Distribution of Bit Patterns in Binary Sequence Generated Over Sub Extension Field
    SJSU ScholarWorks, 2019
    Co-Authors: Ali Md. Arshad, Kodera Yuta, Nogami Yasuyuki, Uehara Satoshi, Morelos-zaragoza Robert
    Abstract:

    The distribution of bit patterns is an important measure to check the randomness of a sequence. The authors of this paper observed this crucial property in a binary sequence which generated by using a primitive polynomial, trace function, and Legendre Symbol defined over the sub extension field. The authors create a new dimension in the sequence generation research area by considering the sub extension field, whereas all our previous works are focused in the prime field. In terms of the distribution of bit patterns property, this research work has notable outcomes more specifically the binary sequence (defined over the sub extension field) holds much better (close to uniform) bit distribution than the previous binary sequence (defined over the prime field). Furthermore, the authors theoretically proved the distribution of bit property in this paper

  • Interleaved sequences of geometric sequences binarized with Legendre Symbol of two types
    'Institute of Electronics Information and Communications Engineers (IEICE)', 2017
    Co-Authors: Tsuchiya Kazuyoshi, Nogami Yasuyuki, Uehara Satoshi
    Abstract:

    A pseudorandom number generator is widely used in cryptography. A cryptographic pseudorandom number generator is required to generate pseudorandom numbers which have good statistical properties as well as unpredictability. An m-sequence is a linear feedback shift register sequence with maximal period over a finite field. M-sequences have good statistical properties, however we must nonlinearize m-sequences for cryptographic purposes. A geometric sequence is a binary sequence given by applying a nonlinear feedforward function to an m-sequence. Nogami, Tada and Uehara proposed a geometric sequence whose nonlinear feedforward function is given by the Legendre Symbol. They showed the geometric sequences have good properties for the period, periodic autocorrelation and linear complexity. However, the geometric sequences do not have the balance property. In this paper, we introduce geometric sequences of two types and show some properties of interleaved sequences of the geometric sequences of two types. These interleaved sequences have the balance property and double the period of the geometric sequences by the interleaved structure. Moreover, we show correlation properties and linear complexity of the interleaved sequences. A key of our observation is that the second type geometric sequence is the complement of the left shift of the first type geometric sequence by half-period positions.Comment: 16 pages, 11 figure

  • Multi-Valued Sequences Generated by Power Residue Symbols over Odd Characteristic Fields
    SJSU ScholarWorks, 2017
    Co-Authors: Nasima Begum, Nogami Yasuyuki, Uehara Satoshi, Morelos-zaragoza Robert
    Abstract:

    This paper proposes a new approach for generating pseudo random multi-valued (including binary-valued) sequences. The approach uses a primitive polynomial over an odd characteristic prime field $\f{p}$, where p is an odd prime number. Then, for the maximum length sequence of vectors generated by the primitive polynomial, the trace function is used for mapping these vectors to scalars as elements in the prime field. Power residue Symbol (Legendre Symbol in binary case) is applied to translate the scalars to k-value scalars, where k is a prime factor of p-1. Finally, a pseudo random k-value sequence is obtained. Some important properties of the resulting multi-valued sequences are shown, such as their period, autocorrelation, and linear complexity together with their proofs and small examples