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2. An elliptic curve having large integral points
- Creator:
- He, Yanfeng and Zhang, Wenpeng
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- elliptic curve, integral point, and Diophantine equation
- Language:
- English
- Description:
- The main purpose of this paper is to prove that the elliptic curve $E\colon y^2=x^3+27x-62$ has only the integral points $(x, y)=(2, 0)$ and $(28844402, \pm 154914585540)$, using elementary number theory methods and some known results on quadratic and quartic Diophantine equations.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
3. On Lehmer's problem and Dedekind sums
- Creator:
- Pan, Xiaowei and Zhang, Wenpeng
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- Lehmer's problem, error term, Dedekind sums, hybrid mean value, and asymptotic formula
- Language:
- English
- Description:
- Let $p$ be an odd prime and $c$ a fixed integer with $(c, p)=1$. For each integer $a$ with $1\le a \leq p-1$, it is clear that there exists one and only one $b$ with $0\leq b \leq p-1$ such that $ab \equiv c $ (mod $p$). Let $N(c, p)$ denote the number of all solutions of the congruence equation $ab \equiv c$ (mod $p$) for $1 \le a$, $b \leq p-1$ in which $a$ and $\overline {b}$ are of opposite parity, where $\overline {b}$ is defined by the congruence equation $b\overline {b}\equiv 1\pmod p$. The main purpose of this paper is to use the properties of Dedekind sums and the mean value theorem for Dirichlet $L$-functions to study the hybrid mean value problem involving $N(c,p)-\frac {1}{2}\phi (p)$ and the Dedekind sums $S(c,p)$, and to establish a sharp asymptotic formula for it.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
4. On the mean value of Dedekind sum weighted by the quadratic Gauss sum
- Creator:
- Wang, Tingting and Zhang, Wenpeng
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- Dedekind sum, quadratic Gauss sum, mean value, and identity
- Language:
- English
- Description:
- Various properties of classical Dedekind sums $S(h, q)$ have been investigated by many authors. For example, Wenpeng Zhang, On the mean values of Dedekind sums, J. Théor. Nombres Bordx, 8 (1996), 429–442, studied the asymptotic behavior of the mean value of Dedekind sums, and H. Rademacher and E. Grosswald, Dedekind Sums, The Carus Mathematical Monographs No. 16, The Mathematical Association of America, Washington, D.C., 1972, studied the related properties. In this paper, we use the algebraic method to study the computational problem of one kind of mean value involving the classical Dedekind sum and the quadratic Gauss sum, and give several exact computational formulae for it.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
5. Some new sums related to D. H. Lehmer problem
- Creator:
- Zhang, Han and Zhang, Wenpeng
- Format:
- print, bez média, and svazek
- Type:
- model:article and TEXT
- Subject:
- Lehmer number, analytic method, trigonometric sums, asymptotic formula, 13, and 51
- Language:
- English
- Description:
- About Lehmer’s number, many people have studied its various properties, and obtained a series of interesting results. In this paper, we consider a generalized Lehmer problem: Let p be a prime, and let N(k; p) denote the number of all 1\leqslant a_{i}\leq p-1 such that a_{1}a_{2}...a_{k}\equiv 1 mod p and 2 | ai + āi + 1, i = 1, 2, ..., k. The main purpose of this paper is using the analytic method, the estimate for character sums and trigonometric sums to study the asymptotic properties of the counting function N(k; p), and give an interesting asymptotic formula for it., Han Zhang, Wenpeng Zhang., and Obsahuje seznam literatury
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
6. Two identities related to Dirichlet character of polynomials
- Creator:
- Yao, Weili and Zhang, Wenpeng
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- Dirichlet character of polynomials, sum analogous to Kloosterman sum, identity, and Gauss sum
- Language:
- English
- Description:
- Let $q$ be a positive integer, $\chi $ denote any Dirichlet character $\mod q$. For any integer $m$ with $(m, q)=1$, we define a sum $C(\chi, k, m; q)$ analogous to high-dimensional Kloosterman sums as follows: $$ C(\chi, k, m; q)=\sum _{a_1=1}^{q}{}' \sum _{a_2=1}^{q}{}' \cdots \sum _{a_k=1}^{q}{}' \chi (a_1+a_2+\cdots +a_k+m\overline {a_1a_2\cdots a_k}), $$ where $a\cdot \overline {a}\equiv 1\bmod q$. The main purpose of this paper is to use elementary methods and properties of Gauss sums to study the computational problem of the absolute value $|C(\chi, k, m; q)|$, and give two interesting identities for it.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public