Do you want to publish a course? Click here

Jacobsthal sums and permutations of biquadratic residues

78   0   0.0 ( 0 )
 Added by Hai-Liang Wu
 Publication date 2020
  fields
and research's language is English




Ask ChatGPT about the research

Let $pequiv1pmod 4$ be a prime. In this paper, with the help of Jacobsthal sums, we study some permutation problems involving biquadratic residues modulo $p$.



rate research

Read More

146 - Hai-Liang Wu , Li-Yuan Wang 2020
In this paper we study products of quadratic residues modulo odd primes and prove some identities involving quadratic residues. For instance, let $p$ be an odd prime. We prove that if $pequiv5pmod8$, then $$prod_{0<x<p/2,(frac{x}{p})=1}xequiv(-1)^{1+r}pmod p,$$ where $(frac{cdot}{p})$ is the Legendre symbol and $r$ is the number of $4$-th power residues modulo $p$ in the interval $(0,p/2)$. Our work involves class number formula, quartic Gauss sums, Stickelbergers congruence and values of Dirichlet L-series at negative integers.
139 - Ce Xu 2020
We define a new kind of classical digamma function, and establish its some fundamental identities. Then we apply the formulas obtained, and extend tools developed by Flajolet and Salvy to study more general Euler type sums. The main results of Flajolet and Salvys paper cite{FS1998} are the immediate corollaries of main results in this paper. Furthermore, we provide some parameterized extensions of Ramanujan-type identities that involve hyperbolic series. Some interesting new consequences and illustrative examples are considered.
240 - Zhong-hua Li 2010
The explicit formulas expressing harmonic sums via alternating Euler sums (colored multiple zeta values) are given, and some explicit evaluations are given as applications.
81 - Weiping Wang , Ce Xu 2020
In this paper, we study the alternating Euler $T$-sums and related sums by using the method of contour integration. We establish the explicit formulas for all linear and quadratic Euler $T$-sums and related sums. Some interesting new consequences and illustrative examples are considered.
193 - Simon Griffiths 2010
A $k$-sum of a set $Asubseteq mathbb{Z}$ is an integer that may be expressed as a sum of $k$ distinct elements of $A$. How large can the ratio of the number of $(k+1)$-sums to the number of $k$-sums be? Writing $kwedge A$ for the set of $k$-sums of $A$ we prove that [ frac{|(k+1)wedge A|}{|kwedge A|}, le , frac{|A|-k}{k+1} ] whenever $|A|ge (k^{2}+7k)/2$. The inequality is tight -- the above ratio being attained when $A$ is a geometric progression. This answers a question of Ruzsa.
comments
Fetching comments Fetching comments
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا