scholarly journals Minimum Number of Additive Tuples in Groups of Prime Order

10.37236/7376 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Ostap Chervak ◽  
Oleg Pikhurko ◽  
Katherine Staden
Keyword(s):  
Fourier Analysis ◽  
Prime Number ◽  
Related Problem ◽  
Prime Order ◽  
Minimum Number ◽  
Modulo P ◽  
Extremal Configuration

For a prime number $p$ and a sequence of integers $a_0,\dots,a_k\in \{0,1,\dots,p\}$, let $s(a_0,\dots,a_k)$ be the minimum number of $(k+1)$-tuples $(x_0,\dots,x_k)\in A_0\times\dots\times A_k$ with $x_0=x_1+\dots + x_k$, over subsets $A_0,\dots,A_k\subseteq\mathbb{Z}_p$ of sizes $a_0,\dots,a_k$ respectively. We observe that an elegant argument of Samotij and Sudakov can be extended to show that there exists an extremal configuration with all sets $A_i$ being intervals of appropriate length. The same conclusion also holds for the related problem, posed by Bajnok, when $a_0=\dots=a_k=:a$ and $A_0=\dots=A_k$, provided $k$ is not equal 1 modulo $p$. Finally, by applying basic Fourier analysis, we show for Bajnok's problem that if $p\geqslant 13$ and $a\in\{3,\dots,p-3\}$ are fixed while $k\equiv 1\pmod p$ tends to infinity, then the extremal configuration alternates between at least two affine non-equivalent sets.   A corrigendum was added March 12, 2019.

scholarly journals Generator of PRN's on the norm group

2021 ◽  
Vol 25 (2(36)) ◽  
pp. 26-39
Author(s):  
P. Fugelo ◽  
S. Varbanets
Keyword(s):  
Prime Number ◽  
Quadratic Field ◽  
Exponential Sum ◽  
Random Numbers ◽  
Gaussian Integers ◽  
New Family ◽  
Serial Test ◽  
Polynomial Representations ◽  
Modulo P ◽  
Norm Group

Let $p$ be a prime number, $d\in\mathds{N}$, $\left(\frac{-d}{p}\right)=-1$, $m>2$, and let $E_m$ denotes the set of of residue classes modulo $p^m$ over the ring of Gaussian integers in imaginary quadratic field $\mathds{Q}(\sqrt{-d})$ with norms which are congruented with 1 modulo $p^m$. In present paper we establish the polynomial representations for real and imagimary parts of the powers of generating element $u+iv\sqrt{d}$ of the cyclic group $E_m$. These representations permit to deduce the ``rooted bounds'' for the exponential sum in Turan-Erd\"{o}s-Koksma inequality. The new family of the sequences of pseudo-random numbers that passes the serial test on pseudorandomness was being buit.

scholarly journals Finite unitary rings with a single subgroup of prime order of the group of units

2020 ◽  
pp. 2150234
Author(s):  
Mostafa Amini ◽  
Mohsen Amiri
Keyword(s):  
Prime Number ◽  
Prime Order ◽  
Finite Ring ◽  
Group Of Units

Let [Formula: see text] be a unitary ring of finite cardinality [Formula: see text], where [Formula: see text] is a prime number and [Formula: see text]. We show that if the group of units of [Formula: see text] has at most one subgroup of order [Formula: see text], then [Formula: see text] where [Formula: see text] is a finite ring of order [Formula: see text] and [Formula: see text] is a ring of cardinality [Formula: see text] which is one of the six explicitly described types.

scholarly journals Bell Numbers Modulo a Prime Number, Traces and Trinomials

10.37236/3532 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Luis H. Gallardo ◽  
Olivier Rahavandrainy
Keyword(s):  
Prime Number ◽  
Finite Fields ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Bell Numbers ◽  
Fixed Integer ◽  
Bell Number ◽  
Modulo P ◽  
Necessary And Sufficient

Given a prime number $p$, we deduce from a formula of Barsky and Benzaghou and from a result of Coulter and Henderson on trinomials over finite fields, a simple necessary and sufficient condition $\beta(n) =k\beta(0)$ in $\mathbb{F}_{p^p}$ in order to resolve the congruence $B(n) \equiv k \pmod{p}$, where $B(n)$ is the $n$-th Bell number, and $k$ is any fixed integer. Several applications of the formula and of the condition are included, in particular we give equivalent forms of the conjecture of Kurepa that $B(p-1)$ is $\neq 1$ modulo $p$.

scholarly journals Vertex-Transitive $q$-Complementary Uniform Hypergraphs

10.37236/587 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Shonda Gosselin
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Prime Order ◽  
Necessary Conditions ◽  
Transitive Permutation Group ◽  
Uniform Hypergraph ◽  
Complete Group ◽  
Large Sets ◽  
Uniform Hypergraphs ◽  
Vertex Transitive

For a positive integer $q$, a $k$-uniform hypergraph $X=(V,E)$ is $q$-complementary if there exists a permutation $\theta$ on $V$ such that the sets $E, E^{\theta}, E^{\theta^2},\ldots, E^{\theta^{q-1}}$ partition the set of $k$-subsets of $V$. The permutation $\theta$ is called a $q$-antimorphism of $X$. The well studied self-complementary uniform hypergraphs are 2-complementary. For an integer $n$ and a prime $p$, let $n_{(p)}=\max\{i:p^i \text{divides} n\}$. In this paper, we prove that a vertex-transitive $q$-complementary $k$-hypergraph of order $n$ exists if and only if $n^{n_{(p)}}\equiv 1 (\bmod q^{\ell+1})$ for every prime number $p$, in the case where $q$ is prime, $k = bq^\ell$ or $k=bq^{\ell}+1$ for a positive integer $b < k$, and $n\equiv 1(\bmod q^{\ell+1})$. We also find necessary conditions on the order of these structures when they are $t$-fold-transitive and $n\equiv t (\bmod q^{\ell+1})$, for $1\leq t < k$, in which case they correspond to large sets of isomorphic $t$-designs. Finally, we use group theoretic results due to Burnside and Zassenhaus to determine the complete group of automorphisms and $q$-antimorphisms of these hypergraphs in the case where they have prime order, and then use this information to write an algorithm to generate all of these objects. This work extends previous, analagous results for vertex-transitive self-complementary uniform hypergraphs due to Muzychuk, Potočnik, Šajna, and the author. These results also extend the previous work of Li and Praeger on decomposing the orbitals of a transitive permutation group.

scholarly journals Serial Computations of Levenshtein Distances

1997 ◽  
Author(s):  
D.S. Hirschberg
Keyword(s):  
Simple Form ◽  
Related Problem ◽  
Minimum Cost ◽  
Longest Common Subsequence ◽  
Distance Measures ◽  
Levenshtein Distance ◽  
String Similarity ◽  
Special Cases ◽  
Common Subsequence ◽  
Minimum Number

In the previous chapters, we discussed problems involving an exact match of string patterns. We now turn to problems involving similar but not necessarily exact pattern matches. There are a number of similarity or distance measures, and many of them are special cases or generalizations of the Levenshtein metric. The problem of evaluating the measure of string similarity has numerous applications, including one arising in the study of the evolution of long molecules such as proteins. In this chapter, we focus on the problem of evaluating a longest common subsequence, which is expressively equivalent to the simple form of the Levenshtein distance. The Levenshtein distance is a metric that measures the similarity of two strings. In its simple form, the Levenshtein distance, D(x , y), between strings x and y is the minimum number of character insertions and/or deletions (indels) required to transform string x into string y. A commonly used generalization of the Levenshtein distance is the minimum cost of transforming x into y when the allowable operations are character insertion, deletion, and substitution, with costs δ(λ , σ), δ(σ, λ), and δ(σ1, σ2) , that are functions of the involved character(s). There are direct correspondences between the Levenshtein distance of two strings, the length of the shortest edit sequence from one string to the other, and the length of the longest common subsequence (LCS) of those strings. If D is the simple Levenshtein distance between two strings having lengths m and n, SES is the length of the shortest edit sequence between the strings, and L is the length of an LCS of the strings, then SES = D and L = (m + n — D)/2. We will focus on the problem of determining the length of an LCS and also on the related problem of recovering an LCS. Another related problem, which will be discussed in Chapter 6, is that of approximate string matching, in which it is desired to locate all positions within string y which begin an approximation to string x containing at most D errors (insertions or deletions).

THE CONTINUING SEARCH FOR LARGE ELITE PRIMES

2009 ◽  
Vol 05 (02) ◽  
pp. 209-218 ◽  
Author(s):  
ALAIN CHAUMONT ◽  
JOHANNES LEICHT ◽  
TOM MÜLLER ◽  
ANDREAS REINHART
Keyword(s):  
Prime Number ◽  
Systematic Search ◽  
Open Problems ◽  
Fermat Numbers ◽  
Quadratic Residues ◽  
Modulo P

A prime number p is called elite if only finitely many Fermat numbers 22n + 1 are quadratic residues modulo p. So far, all 21 elite primes less than 250 billion were known, together with 24 larger items. We completed the systematic search up to the range of 2.5 · 1012 finding six more such numbers. Moreover, 42 new elites larger than this bound were found, among which the largest has 374 596 decimal digits. A survey on the knowledge about elite primes together with some open problems and conjectures are presented.

scholarly journals An Efficient Approach to Point-Counting on Elliptic Curves from a Prominent Family over the Prime Field Fp

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1431
Author(s):  
Yuri Borissov ◽  
Miroslav Markov
Keyword(s):  
Elliptic Curves ◽  
Prime Number ◽  
Explicit Formula ◽  
Point Counting ◽  
Prime Field ◽  
Algorithmic Solution ◽  
The Family ◽  
Order Of Magnitude ◽  
Modulo P ◽  
Quantities Of Interest

Here, we elaborate an approach for determining the number of points on elliptic curves from the family Ep={Ea:y2=x3+a(modp),a≠0}, where p is a prime number >3. The essence of this approach consists in combining the well-known Hasse bound with an explicit formula for the quantities of interest-reduced modulo p. It allows to advance an efficient technique to compute the six cardinalities associated with the family Ep, for p≡1(mod3), whose complexity is O˜(log2p), thus improving the best-known algorithmic solution with almost an order of magnitude.

scholarly journals On a Generalization of a Lucas’ Result and an Application to the 4-Pascal’s Triangle

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 288
Author(s):  
Atsushi Yamagami ◽  
Kazuki Taniguchi
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Prime Number ◽  
Special Issue ◽  
Pascal’S Triangle ◽  
Mersenne Number ◽  
Modulo P ◽  
Pascal's Triangle ◽  
Positive Integers ◽  
Symmetric Property

The Pascal’s triangle is generalized to “the k-Pascal’s triangle” with any integer k ≥ 2 . Let p be any prime number. In this article, we prove that for any positive integers n and e, the n-th row in the p e -Pascal’s triangle consists of integers which are congruent to 1 modulo p if and only if n is of the form p e m − 1 p e − 1 with some integer m ≥ 1 . This is a generalization of a Lucas’ result asserting that the n-th row in the (2-)Pascal’s triangle consists of odd integers if and only if n is a Mersenne number. As an application, we then see that there exists no row in the 4-Pascal’s triangle consisting of integers which are congruent to 1 modulo 4 except the first row. In this application, we use the congruence ( x + 1 ) p e ≡ ( x p + 1 ) p e − 1 ( mod p e ) of binomial expansions which we could prove for any prime number p and any positive integer e. We think that this article is fit for the Special Issue “Number Theory and Symmetry,” since we prove a symmetric property on the 4-Pascal’s triangle by means of a number-theoretical property of binomial expansions.

scholarly journals NP-Completeness Results for Minimum Planar Spanners

1998 ◽  
Vol Vol. 3 no. 1 ◽  
Author(s):  
Ulrik Brandes ◽  
Dagmar Handke
Keyword(s):  
Related Problem ◽  
Interesting Case ◽  
Edge Weight ◽  
Weighted Graphs ◽  
Np Completeness ◽  
Spanning Subgraph ◽  
Fixed Parameter ◽  
Minimum Number ◽  
International Audience ◽  
Np Hardness

International audience For any fixed parameter t greater or equal to 1, a \emph t-spanner of a graph G is a spanning subgraph in which the distance between every pair of vertices is at most t times their distance in G. A \emph minimum t-spanner is a t-spanner with minimum total edge weight or, in unweighted graphs, minimum number of edges. In this paper, we prove the NP-hardness of finding minimum t-spanners for planar weighted graphs and digraphs if t greater or equal to 3, and for planar unweighted graphs and digraphs if t greater or equal to 5. We thus extend results on that problem to the interesting case where the instances are known to be planar. We also introduce the related problem of finding minimum \emphplanar t-spanners and establish its NP-hardness for similar fixed values of t.

scholarly journals A lower bound concerning subset sums which do not cover all the residues modulo $p$.

2005 ◽  
Vol Volume 28 ◽  
Author(s):  
Jean-Marc Deshouillers
Keyword(s):  
Lower Bound ◽  
Prime Number ◽  
Modulo P ◽  
Subset Sums ◽  
International Audience

International audience Let $c>\sqrt{2}$ and let $p$ be a prime number. J-M. Deshouillers and G. A. Freiman proved that a subset $\mathcal A$ of $\mathbb{Z}/p\mathbb{Z}$, with cardinality larger than $c\sqrt{p}$ and such that its subset sums do not cover $\mathbb{Z}/p\mathbb{Z}$ has an isomorphic image which is rather concentrated; more precisely, there exists $s$ prime to $p$ such that $$\sum_{a\in\mathcal A}\Vert\frac{as}{p}\Vert < 1+O(p^{-1/4}\ln p),$$ where the constant implied in the ``O'' symbol depends on $c$ at most. We show here that there exist a $K$ depending on $c$ at most, and such sets $\mathcal A$, such that for all $s$ prime to $p$ one has $$ \sum_{a\in\mathcal A}\Vert\frac{as}{p}\Vert>1+Kp^{-1/2}.$$

Sign in / Sign up

Export Citation Format

Share Document