Pingzhi Yuan

Pingzhi Yuan
South China Normal University · School of Mathematics

Doctor of Philosophy

About

195
Publications
15,714
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1,259
Citations
Additional affiliations
January 2009 - present
South China Normal University
Position
  • Professor (Full)
January 2009 - January 2016
South China Normal University
Position
  • Professor (Full)
January 2009 - July 2019
South China Normal University
Position
  • Master's Student

Publications

Publications (195)
Preprint
Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we determine all permutation trinomials over $\mathbb{F}_{2^m}$ in Zieve's paper. We prove a conjecture proposed by Gupta and Sharma and obtain some new permutation trinomials over $\mathbb{F}_{2^m}$....
Article
In this paper, we study nonsingular splittings of cyclic groups. We introduce two new notations “direct logarithm” and” special direct KM-logarithm”. Using Kummer and Mills' results on characters and a result from the factorization of abelian groups, we prove that if there is a prime q such that M splits Zq, then there are infinitely many primes p...
Article
Full-text available
For a prime p and positive integers m, n, let Fq be a finite field with q=pm elements and Fqn be an extension of Fq. Let h(x) be a polynomial over Fqn satisfying the following conditions: (i) Trmnm(x)∘h(x)=τ(x)∘Trmnm(x); (ii) For any s∈Fq, h(x) is injective on Trmnm(x)-1(s), where τ(x) is a polynomial over Fq. For b,c∈Fq,δ∈Fqn, and positive integer...
Preprint
In this paper, we prove that every PP is an AGW-PP. We also extend the result of Wan and Lidl to other permutation polynomials over finite fields and determine their group structure. Moreover, we provide a new general method to find the compositional inverses of all PPs, some new PPs and their compositional inverses are given.
Article
Let $G$ be a finite abelian group. We say that $M$ and $S$ form a splitting of $G$ if every nonzero element $g$ of $G$ has a unique representation of the form $g = ms$ with $m\in M$ and $s\in S$, while 0 has no such representation.The splitting is called nonsingular if $\gcd(|G|, a) = 1$ for any $a\in M$. The splitting is called purely singular if...
Article
Let s be a prime power and \( {{{\mathbb {F}}}}_q\) be a finite field with s elements. In this paper, we employ the AGW criterion to investigate the permutation behavior of some polynomials of the form $$\begin{aligned} b(x^q+ax+\delta )^{1+\frac{i(q^2-1)}{d}}+c(x^q+ax+\delta )^{1+\frac{j(q^2-1)}{d}}+L(x) \end{aligned}$$over \( {{{\mathbb {F}}}}_{q...
Article
MSC: 11A05 11B83 Keywords: Least common multiple Reciprocal Exact upper bound Highly composite number Divisor sequence r-optimal sequence a b s t r a c t Let r be a positive number with r ≥ 2 and let A = { a i } ∞ i =1 be an arbitrarily given strictly increasing sequence of positive integers. Let S A r := ∞ i =1 1 [ a i ,a i +1 , ... ,a i + r−1 ]....
Preprint
In this paper, we present two methods to obtain the compositional inverses of AGW-PPs. We improve some known results in this topic.
Article
Full-text available
We proved that for any positive integer $a>2$ and $k>1$, the congruence equation $a^{n-k}\equiv1\pmod{n}$ has infinitely many positive integer solution $n$.
Article
In this paper, we present a new method to obtain the compositional inverses of AGW-PPs. We improve some known results in this topic.
Article
An element [Formula: see text] of a ring [Formula: see text] is called a quasi-idempotent if [Formula: see text] for some central unit [Formula: see text] of [Formula: see text], or equivalently, [Formula: see text], where [Formula: see text] is a central unit and [Formula: see text] is an idempotent of [Formula: see text]. A ring [Formula: see tex...
Article
Full-text available
Let $Z$ and $N$ be the set of all integers and positive integers, respectively. $M_m (Z)$ be the set of $m\times m$ matrix over $Z$ where $ m\in N$. In this paper, by using the result of Fermat’s Last Theorem, we show that the following second-order matrix equation has only trivial solutions: $$ X^n + Y^ n = \lambda^n I \quad (\almbda\in Z, \lambda...
Preprint
In 2012, T. Miyazaki and A. Togb\'{e} gave all of the solutions of the Diophantine equations $(2am-1)^x+(2m)^y=(2am+1)^z$ and $b^x+2^y=(b+2)^z$ in positive integers $x,y,z,$ $a>1$ and $b\ge 5$ odd. In this paper, we propose a similar problem (which we call the shuffle variant of a Diophantine equation of Miyazaki and Togb\'{e}). Here we first prove...
Article
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4 8 卷第 4 期 2 0 0 5 年 7月 数 学 学 报 A C T A M A T H E M A T IC A S IN IC A V 6 1. 4 8 , N o 4 J u ly , 2 0 0 5 文章编号 : 0 5 8 3 一 1 4 3 1 (2 0 0 5) 04 一 0 70 7 一 0 8 文献标识码 : A L e h m e r 序列 中的平方数与平方类 罗家贵 海南 大 学数学系 海 口 5 7 0 2 2 8-E m a i l : j g 一 u o @ t o m. e o m 关键词 二 阶序列 ; L h e m e r 序列 ; a J c o ib 符 号 M R (2 0 0 0) 主题分类 1 I B 3 9 , 一z D 6 1 中图分...
Preprint
Let G be a finite abelian group, M a set of integers and S a subset of G. We say that M and S form a splitting of G if every nonzero element g of G has a unique representation of the form g = ms with m ∈ M and s ∈ S, while 0 has no such representation. The splitting is called purely singular if for each prime divisor p of |G|, there is at least one...
Article
In this paper, we consider a family of elliptic curves over the rational numbers which arise from a well-known problem of Zagier. The main result is to prove a sharp upper bound of one nontrivial integral point on any curve in this family. This result generalizes some results which appear in the literature, and simplifies the proof for those previo...
Article
In this note, we show that each positive rational number can be written uniquely as $\varphi(m^2)/\varphi(n^2)$, where $m, \, n\in\mathbb{N}$, with some natural restrictions on $\gcd(m, n)$.
Preprint
Full-text available
We say that $M$ and $S$ form a \textsl{splitting} of $G$ if every nonzero element $g$ of $G$ has a unique representation of the form $g=ms$ with $m\in M$ and $s\in S$, while $0$ has no such representation. The splitting is called {\it nonsingular} if $\gcd(|G|, a) = 1$ for any $a\in M$. In this paper, we focus our study on nonsingular splittings of...
Article
Let G be a finite abelian group, M a set of integers and S a subset of G. We say that M and S form a splitting of G if every nonzero element g of G has a unique representation of the form g=ms with m∈ M and s∈S, while 0 has no such representation. The splitting is called purely singular if for each prime divisor p of |G|, there is at least one elem...
Article
In 2012, T. Miyazaki and A. Togbé gave all of the solutions of the Diophantine equations (2am − 1) x + (2m) y = (2am + 1) z and b x + 2 y = (b + 2) z in positive integers x, y, z, a > 1 and b ≥ 5 odd. In this paper, we propose a similar problem (which we call the shuffle variant of a Diophantine equation of Miyazaki and Togbé). Here we first prove...
Article
In 2012, Tian and Zhou conjectured that a flag-transitive and point-primitive automorphism group of a symmetric $(v,k,\lambda)$ design must be an affine or almost simple group. In this paper, we study this conjecture and prove that if $k\leq 10^3$ and $G\leq Aut(\mathcal{D})$ is flag-transitive and point-primitive, then $G$ is affine or almost simp...
Article
Full-text available
Let (Equation) and (Equation) be positive integers with (Equation) . In this paper, we show that every positive rational number can be written as the form (Equation) , where m,n∈N if and only if (Equation) or (Equation) . Moreover, if (Equation) , then the proper representation of such representation is unique.
Preprint
Full-text available
Suppose that $n$ is a positive integer. In this paper, we show that the exponential Diophantine equation $$(n-1)^{x}+(n+2)^{y}=n^{z},\ n\geq 2,\ xyz\neq 0$$ has only the positive integer solutions $(n,x,y,z)=(3,2,1,2), (3,1,2,3)$. The main tools on the proofs are Baker's theory and Bilu-Hanrot-Voutier's result on primitive divisors of Lucas numbers...
Preprint
Let $G$ be a finite abelian group. We say that $M$ and $S$ form a \textsl{splitting} of $G$ if every nonzero element $g$ of $G$ has a unique representation of the form $g=ms$ with $m\in M$ and $s\in S$, while $0$ has no such representation. The splitting is called \textit{purely singular} if for each prime divisor $p$ of $|G|$, there is at least on...
Article
Given integers $k_1, k_2$ with $0\le k_1<k_2$, the determination of all positive integers $q$ for which there exists a perfect splitter $B[-k_1, k_2](q)$ set is a wide open question in general. In this paper, we obtain new necessary and sufficient conditions for an odd prime $p$ such that there exists a nonsingular perfect $B[-1,3](p)$ set. We also...
Article
Full-text available
Suppose that $n$ is a positive integer. In this paper, we show that the exponential Diophantine equation $$(n-1)^{x}+(n+2)^{y}=n^{z}, \, \, n\geq 2, \, \, xyz\neq 0$$ has only the positive integer solutions $(n,x,y,z)=(3,2,1,2), (3,1,2,3)$. The main tool on the proofs is Bilu-Hanrot-Voutier's result on primitive divisors of Lucas numbers.
Article
Full-text available
We prove that the equation entitled has only the trivial positive integer solution $(x, y, z)=(1, 1, 2)$.
Preprint
Given integers $k_1, k_2$ with $0\le k_1<k_2$, the determinations of all positive integers $q$ for which there exists a perfect Splitter $B[-k_1, k_2](q)$ set is a wide open question in general. In this paper, we obtain new necessary and sufficient conditions for an odd prime $p$ such that there exists a nonsingular perfect $B[-1,3](p)$ set. We als...
Preprint
In 1956, Je$\acute{s}$manowicz conjectured that, for positive integers $m$ and $n$ with $m>n, \, \gcd(m,\, n)=1$ and $m\not\equiv n\pmod{2}$, the exponential Diophantine equation $(m^2-n^2)^x+(2mn)^y=(m^2+n^2)^z$ has only the positive integer solution $(x,\,y,\, z)=(2,\,2,\,2)$. Recently, Ma and Chen \cite{MC17} proved the conjecture if $4\not|mn$...
Article
Full-text available
We show that an order m dimension 2 tensor is primitive if and only if its majorization matrix is primitive, and then we obtain the characterization of order m dimension 2 strongly primitive tensors and the bound of the strongly primitive degree. Furthermore, we study the properties of strongly primitive tensors with n≥3 and propose some problems f...
Article
Full-text available
Let \(m \ge 5\) be an odd integer. For \(d=2^m+2^{(m+1)/2}+1\) or \(d=2^{m+1}+3\), Blondeau et al. conjectured that the power function \(F_d=x^d\) over \(\mathrm {GF}(2^{2m})\) is differentially 8-uniform in which all values \(0, \, 2, \, 4,\, 6,\, 8\) appear. In this paper, we confirm this conjecture and compute the differential spectrum of \(F_d\...
Article
Full-text available
Jeśmanowicz [9] conjectured that, for positive integers m and n with m > n, gcd(m,n) = 1 and \({m\not\equiv n\pmod{2}}\), the exponential Diophantine equation \({(m^2-n^2)^x+(2mn)^y=(m^2+n^2)^z}\) has only the positive integer solution (x, y, z) = (2, 2, 2). We prove the conjecture for \({2 \| mn}\) and m + n has a prime factor p with \({p\not\equi...
Article
Full-text available
Text: Let k,l,a,b,c be positive integers such that gcd (ka,lb)=1, min (a,b,c)>1, a≠3, b≠3 and 2 c. In this paper, we prove that there are at most four solutions in positive integers (x,y,z) to the equation kax+lby=cz and at most two solutions when 2 (u(l/k)), where u(m) is the least positive integer t with mt≡1(modc). Video: For a video summary of...
Article
For positive integers N and M, the general hypergeometric Cauchy polynomialscM,N,n(z) (M, N ≥ 1; n ≥ 0) are defined by $$\frac{1}{(1+t)^z} \frac{1}{{}_2F_1(M,N;N+1;-t)}=\sum_{n=0}^\infty c_{M,N,n}(z)\, \frac{t^n}{n!}\,, $$where \({{}_2 F_1(a,b;c;z)}\) is the Gauss hypergeometric function. When M = N = 1, cn = c1,1,n are the classical Cauchy numbers...
Article
Full-text available
Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we determine all permutation trinomials over F2min Zieve's paper [30]. We prove a conjecture proposed by Gupta and Sharma in [8] and obtain some new permutation trinomials over F2m. Finally, we show t...
Article
Full-text available
In this paper, we show that an order $m$ dimension 2 tensor is primitive if and only if its majorization matrix is primitive, and then we obtain the characterization of order $m$ dimension 2 strongly primitive tensors and the bound of the strongly primitive degree. Furthermore, we study the properties of strongly primitive tensors with $n\geq 3$, a...
Article
Full-text available
Permutation polynomials (PPs) of the form \((x^{q} -x + c)^{\frac{q^2 -1}{3}+1} +x\) over F_{q^2} were presented by Li et al. (Finite Fields Appl 22:16–23, 2013). More recently, we have constructed PPs of the form \((x^{q} + bx + c)^{\frac{q^2 -1}{d}+1} -bx\) over F_{q^2} , where \(d=2, 3, 4, 6\) (Yuan and Zheng in Finite Fields Appl 35:215–230, 20...
Article
Full-text available
A ring with involution * is called *-clean if each of its elements is the sum of a unit and a projection (*-invariant idempotent). Recently, Gao, Chen, and Li obtained necessary and sufficient conditions for RG to be *-clean, where R is a commutative local ring and G is one of C3, C4, S3, and Q8. Most recently, Li, Yuan, and Parmenter gave a comple...
Article
Full-text available
In this paper, we mainly discuss the characterization of a class of arithmetic functions $f: N \rightarrow C$ such that $f(u^{2}+kv^2)=f^{2}(u)+kf^{2}(v)$ $(k, u, v \in N)$. We obtain a characterization with given condition, propose a conjecture and show the result holds for $k \in \{2, 3, 4, 5 \}$.
Article
For n ∈ N, we denote by π(n) the set of prime divisors of n. Let G be a finite group. Denote by IBrp(G) and cl(Gp′) the set of irreducible p-Brauer characters and the set of p-regular conjugacy classes of G, respectively. Set ρp(G) = {q prime: q|ϕ(1), ϕ ∈IBrp(G)}, (Formula presented.) , σp(G) = max{|π(ϕ(1))|: ϕ ∈IBrp(G)}, and (Formula presented.)....
Article
In this paper, we obtain the sharp upper and lower bounds for the spectral radius of a nonnegative irreducible matrix. We also apply these bounds to various matrices associated with a graph or a digraph, obtain some new results or known results about various spectral radii, including the adjacency spectral radius, the signless Laplacian spectral ra...
Article
Full-text available
For any co-prime positive integers a and b with \({a \equiv -1}\) (mod b), we determine all positive solutions \({(x, y, z)}\) of the Diophantine equation in the title.
Article
Text. Let f (n) be a multiplicative function such that f does not vanish at some prime po. In this paper, it is proved that, for any given integer no with 1 <= n(0) <= 10(6), if f (p + q + n(0)) = f (p) + f (q) + f (n(0)) for all primes p and q, then f must be the identity function: f (n) = n for all integers n >= 1. If a variation of Goldbach's co...
Article
Full-text available
It is a hard problem to find the inverse of a nontrivial class of permutation polynomials of finite fields. In this paper the piecewise method is employed to construct the inverses of permutation polynomials, although piecewise constructing permutation polynomials is not a new idea. A formula for the inverses of some permutation polynomials of fini...
Article
Full-text available
Four recursive constructions of permutation polynomials over $\gf(q^2)$ with those over $\gf(q)$ are developed and applied to a few famous classes of permutation polynomials. They produce infinitely many new permutation polynomials over $\gf(q^{2^\ell})$ for any positive integer $\ell$ with any given permutation polynomial over $\gf(q)$. A generic...
Article
Let G be a finite Abelian group of order |G| = n, and let S = g1.….gn−1 be a sequence over G such that all nonempty zero-sum subsequences of S have the same length. In this paper, we completely determine the structure of these sequences.
Article
Full-text available
Permutation polynomials (PPs) of the form $(x^{q} -x + c)^{\frac{q^2 -1}{3}+1} +x$ over $\F_{q^2}$ were presented by Li, Helleseth and Tang [Finite Fields Appl. 22 (2013) 16--23]. More recently, we have constructed PPs of the form $(x^{q} +bx + c)^{\frac{q^2 -1}{d}+1} -bx$ over $\F_{q^2}$, where $d=2, 3, 4, 6$ [Finite Fields Appl. 35 (2015) 215--23...
Article
Full-text available
Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we present some classes of explicit permutation polynomials of the forms similar to , . The results here generalize the similar results obtained in [5] and [15] greatly.
Article
In this paper, we show that the primitive degree set of nonnegative primitive tensors with order and dimension n is , which implies that the results of the case (the case of tensors) is totally different from the case (the case of matrices), and we also propose some open problems for further research.
Article
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In an additively written abelian group, a sequence is called zero-sum free if each of its nonempty subsequences has sum different from the zero element of the group. In this paper, we consider the structure of long zero-sum free sequences and n-zero-sum free sequences over finite cyclic groups Zn. Among which, we determine the structure of the long...
Article
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Let G be an additively written finite cyclic group of order n and let S be a minimal zero-sum sequence with elements of G, i.e. the sum of elements of S is zero, but no proper nontrivial subsequence of S has sum zero. S is called unsplittable if there do not exist an element g in S and two elements x, y in G such that g = x + y and the new sequence...
Article
Full-text available
In this paper, we obtain the sharp upper and lower bounds for the spectral radius of a nonnegative irreducible matrix. We also apply these bounds to various matrices associated with a graph or a digraph, obtain some new results or known results about various spectral radii, including the adjacency spectral radius, the signless Laplacian spectral ra...
Article
Full-text available
Let $H$ be a Krull monoid with class group $G$ such that every class contains a prime divisor (for example, rings of integers in algebraic number fields or holomorphy rings in algebraic function fields). For $k \in \mathbb N$, let $\mathcal U_k (H)$ denote the set of all $m \in \mathbb N$ with the following property: There exist atoms $u_1, ..., u_...
Article
A ring with involution * is called *-clean if each of its elements is the sum of a unit and a projection. Clearly a *-clean ring is clean. Vaš asked whether there exists a clean ring with involution * that is not *-clean. In a recent paper, Gao, Chen and the first author investigated when a group ring RG with classical involution * is *-clean and o...
Article
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In this paper, we consider infinite sums derived from the reciprocals of the generalized Fibonacci numbers. We obtain some new and interesting identities for the generalized Fibonacci numbers.
Article
Let q be a power of 2, k a positive integer and let . In this paper, we present a number of classes of explicit permutation polynomials on , which are of the form , where , , and are linearized polynomials. We also point out an application of these permutation polynomials in combinatorics, cryptography and sequences.
Article
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In this paper, we show that the exponent set of nonnegative primitive tensors with order m(\geq 3) and dimension n is {1,2,\ldots, (n-1)^2+1}; and propose some open problems for further research.
Article
A conjecture proposed by Jeśmanowicz on Pythagorean triples states that for any fixed primitive Pythagorean triple (a,b,c)(a,b,c) such that a2+b2=c2a2+b2=c2, the Diophantine equation ax+by=czax+by=cz has only the trivial solution in positive integers x,yx,y and z. In this paper we establish the conjecture for the case where b is even and either a o...
Article
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Let H be a Krull monoid with finite class group G such that every class contains a prime divisor. The monotone catenary degree c mon (H) of H is the smallest integer m with the following property: for each ${a \in H}$ and each two factorizations z, z′ of a with length |z| ≤ |z′|, there exist factorizations z = z 0, ... ,z k = z′ of a with increas...
Article
We obtain all positive integer solutions (m 1,m 2, a, b) with a > b, gcd(a, b) = 1 to the system of Diophantine equations \(km_1^2 - la^{t_1 } b^{t_2 } a^{2r} = C_1\), \(km_2^2 - la^{t_1 } b^{t_2 } b^{2r} = C_2\) with C 1,C 2 ∈ {−1, 1,−2, 2, −4, 4}, and k, l, t 1, t 2, r ∈ ℤ such that k > 0,l > 0, r > 0, t 1 > 0, t 2 ⩾ 0, gcd(k, l) = 1, and k is sq...
Article
Recently, Song and Qi extended the concept of P, P 0 and B matrices to P, P 0 , B and B 0 tensors, obtained some properties about these tensors, and proposed many questions for further research. In this paper, we answer three questions mentioned as above and obtain further results about P, P 0 , B and B 0 tensors.
Article
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Let H be an abelian group written additively and k be a positive integer. Let G(H, k) denote the digraph whose set of vertices is just H, and there exists a directed edge from a vertex a to a vertex b if b = ka. In this paper we give a necessary and sufficient condition for G(H, k1) ≃ G(H, k 2). We also discuss the problem when G(H1, k) is isomorph...
Article
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In this paper, we present a necessary and sufficient condition for a nonnegative tensor to be a primitive one, show that the exponent set of nonnegative primitive tensors with order $m(\ge n)$ and dimension $n$ is $\{k| 1\le k\le (n-1)^2+1\}. $
Article
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Permutation polynomials have been a subject of study for a long time and have applications in many areas of science and engineering. However, only a small number of specific classes of permutation polynomials are described in the literature so far. In this paper we present a number of permutation trinomials over finite fields, which are of differen...
Article
A natural number n is called k-perfect if σ(n)=kn. In this paper, we show that for any integers r≥2 and k≥2, the number of odd k-perfect numbers n with ω(n)≤r is bounded by ⌊4 r log 3 2⌋+r r∑ i=1 r ⌊kr/2⌋ i, which is less than 4 r 2 when r is large enough.
Article
Let q be a power of 2, k a positive integer and let Sk=x+xq+⋯+xqk−1∈Fq[x]Sk=x+xq+⋯+xqk−1∈Fq[x]. In this paper, we present a number of classes of explicit permutation polynomials on Fq3kFq3k, which are of the form L(x)+S2ka+S2kb, where a∈{1,qk,q2k}a∈{1,qk,q2k}, b∉{1,qk,q2k}b∉{1,qk,q2k}, and L(x)L(x) are linearized polynomials. We also point out an a...
Chapter
Let 1 ≦ a1 ≦ a2 ≦ … ≦ as be integers with ∑si=1 1/ai ≧ n + 9/31. In this paper, we prove that this sum can be decomposed into n parts so that all partial sums are greater than or equal to 1.
Chapter
For any given positive integer l, we prove that there are only finitely many integers k such that the Diophantine equation x²-kxy+y²+lx = 0 has an infinite number of positive integer solutions (x, y). Moreover, we determine all integers k such that the Diophantine equation x²-kxy+y²+lx = 0, 1 ≤ l ≤ 33, has an infinite number of positive integer sol...
Article
In this paper, we prove: Let A be a nonnegative primitive tensor with order m and dimension n. Then its primitive degree R(A)\leq (n-1)^2+1, and the upper bound is sharp. This con?rms a conjecture of Shao [7].
Article
Let $\mathbb{P}_n$ be the set of all matrices which have the same zero patterns with some permutation matrix of order $n$. In this paper, we prove the following result: Let $\mathbb{I}$ be the unit tensor of order $m\ge3$ and dimension $n\ge2$. Suppose that $P$ and $Q$ are two matrices with $P\mathbb{I}Q=\mathbb{I}$, then $P,Q\in \mathbb{P}_n$. Thi...
Conference Paper
In this paper, we propose a scheme about blind signature based on quartic residues, and present a formal proof for its security. Because the security of the scheme is based on the factorization problem of large integer, we discuss the relationship between them both in detail.
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Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we develop general theorems on permutation polynomials over finite fields. As a demonstration of the theorems, we present a number of classes of explicit permutation polynomials on $\gf_q$.
Article
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For any given positive integer l, we prove that there are only finitely many integers k such that the Diophantine equation x 2 -kxy+y 2 +lx=0 has an infinite number of positive integer solutions (x,y). Moreover, we determine all integers k such that the Diophantine equation x 2 -kxy+y 2 +lx=0, 1≤l≤33, has an infinite number of positive integer solu...
Article
Let $x>1, y>1,z>1$ be positive integers of the Diophantine equation $x^y+y^x=z^z$. Using the lower bound of the linear form in $p$-adic logarithm, it is proved that $x, y, z$ are coprime integers and $z<2.8\times10^9$.
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Let 1≤a 1 ≤a 2 ≤⋯≤a s be integers with ∑ i=1 s 1/a i ≥n+9/31. In this paper, we prove that this sum can be decomposed into n parts so that all partial sums are greater than or equal to 1.
Article
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Let a,b,c be integers. In this paper, we prove the integer solutions of the equation ax y +by z +cz x =0 satisfy max{|x|,|y|,|z|}≤2max{a,b,c} when a,b,c are odd positive integers, and when a=b=1,c=-1, the positive integer solutions of the equation satisfy max{x,y,z}<exp(exp(exp(5))).
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Let a and b be distinct positive integers. In this paper, we will present some new results on the positive integer solutions (n; x) of the equation of the title.
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For any positive integers nn and kk, let G(n,k)G(n,k) denote the digraph whose set of vertices is H={0,1,2,…,n−1}H={0,1,2,…,n−1} and there is a directed edge from a∈Ha∈H to b∈Hb∈H if ak≡b(modn). The digraph G(n,k)G(n,k) is called symmetric of order MM if its set of connected components can be partitioned into subsets of size MM with each subset con...
Article
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Authors’ abstract: Recently, Tu and Deng [A conjecture on binary string and its application on constructing Boolean Functions of optimal algebraic immunity. Designs, Codes and Cryptography 60, 1 - 14 (2010; Zbl 1226.94013)] obtained two classes of Boolean functions with nice properties based on a combinatorial conjecture about binary strings. In th...
Article
Applying results on linear forms in p-adic logarithms, we prove that if (x,y,z) is a positive integer solution to the equation x y -y x =c z with gcd(x,y)=1 then (x,y,z)=(2,1,k),(3,2,k),k≥1 if c=1, and either (x,y,z)=(c k +1,1,k),k≥1 or 2≤x<y≤max{1·5×10 10 ,c} if c≥2.
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Applying results on linear forms in p-adic logarithms, we prove that if (x,y,z) is a positive integer solution to the equation xy-yx=cz with gcd(x,y)=1 then (x,y,z)=(2,1,k), (3,2,k), k≥1 if c=1, and either (x,y,z)=(ck+1,1,k), k≥1 or 2≤x<y≤max{1.5×1010,c} if c≥2.
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We obtain all solutions of the equation $\frac{ax^{n+2l}+c}{abt^{2}x^{n}+c} = by^{2}$ with c∈{±1,±2,±4}.
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Recently, Yuan and Li [Zbl 1216.11038] considered a variant y 2 =px(Ax 2 -2) of Cassels’ equation y 2 =3x(x 2 +2) [Cassels, Glasg. Math. J. 27, 11–18 (1985; Zbl 0576.10010)]. They proved that the equation has at most five solutions in positive integers (x,y). In this note, we improve Yuan-Li’s result by showing that for any prime p and any odd posi...
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Let G be a cyclic group of order n. and let S is an element of F(G) be a zero-sum sequence of length vertical bar S vertical bar >= 2[n/2] + 2. Suppose that S can be decomposed into a product of at most two minimal zero-sum sequences. Then there exists some g E G such that S = (n(1)g) . (n(2)g) ..... (n(vertical bar s vertical bar)g), where n(i) is...
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Using a lemma proved by Akbary, Ghioca, and Wang, we derive several theorems on permutation polynomials over finite fields. These theorems give not only a unified treatment of some earlier constructions of permutation polynomials, but also new specific permutation polynomials over Fq. A number of earlier theorems and constructions of permutation po...
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In 1992, Ma made a conjecture related to Pell equations. In this paper, we use Störmer's theorem and related results on Pell equations to prove some particular cases of Ma's conjecture.
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Kai Zhou (2008) [8] gave an explicit representation of the class of linear permutation polynomials and computed the number of them. In this paper, we give a simple proof of the above results.
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Let F(x,y) be an irreducible binary form of degree r>=3, with rational integral coefficients. Let Nr be the number of solutions of the equation |F(x,y)| = 1, then using the methods developed by Bombieri, Schmidt and Stewart, we prove that Nr=24 and that Nr=100. For 4

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