scholarly journals On the difference of values of the kernel function at consecutive integers

2003 ◽  
Vol 2003 (67) ◽  
pp. 4249-4262
Author(s):  
Jean-Marie De Koninck ◽  
Florian Luca
Keyword(s):  
Positive Integer ◽  
Kernel Function ◽  
Large Family ◽  
Fixed Integer ◽  
Consecutive Integers ◽  
The Difference

For each positive integern, setγ(n)=Πp|np. Given a fixed integerk≠±1, we establish that if theABC-conjecture holds, then the equationγ(n+1)−γ(n)=khas only finitely many solutions. In the particular casesk=±1, we provide a large family of solutions for each of the corresponding equations.

scholarly journals A lower bound for a remainder term associated with the sum of digits function

1991 ◽  
Vol 34 (1) ◽  
pp. 121-142 ◽  
Author(s):  
D. M. E. Foster
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Upper Bound ◽  
Remainder Term ◽  
Fixed Integer ◽  
Sum Of Digits ◽  
Sum Of Digits Function ◽  
The Difference

For a fixed integer q≧2, every positive integer k = Σr≧0ar(q, k)qr where each ar(q, k)∈{0,1,2,…, q−1}. The sum of digits function α(q, k) Σr≧0ar(q, k) behaves rather erratically but on averaging has a uniform behaviour. In particular if , where n>1, then it is well known that A(q, n)∼½((q − 1)/log q)n logn as n → ∞. For odd values of q, a lower bound is now obtained for the difference 2S(q, n) = A(q, n)−½(q − 1))[log n/log q, where [log n/log q] denotes the greatest integer ≦log n /log q. This complements an upper bound already found.

scholarly journals Averaging the sum of digits function to an even base

1992 ◽  
Vol 35 (3) ◽  
pp. 449-455 ◽  
Author(s):  
D. M. E. Foster
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Fixed Integer ◽  
Sum Of Digits ◽  
Sum Of Digits Function ◽  
The Difference

For a fixed integer q≧2, every positive integer where each ar(q, k) ∈ {0, 1, 2, …, q–1}. The sum of digits function α(q, k) = behaves rather erratically but on averaging has a uniform behaviour. In particular if A(q, n) = , where n > 1, then it is well known that A(q, n)∼½ ((q – 1)/log q) n log n as n→∞. For even values of q, a lower bound is now given for the difference ½S(q, n) = A(q, n)–½(q–1)[logn/logq] n, where [log n/log q] denotes the greatest integer ≦ log n/log q, complementing an earlier result for odd values of q.

scholarly journals On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1813
Author(s):  
S. Subburam ◽  
Lewis Nkenyereye ◽  
N. Anbazhagan ◽  
S. Amutha ◽  
M. Kameswari ◽  
...  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Exponential Diophantine Equation ◽  
Research Article ◽  
All Solutions ◽  
Sum Of Products ◽  
Consecutive Integers ◽  
Sums Of Products ◽  
Positive Integers

Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n= 19,736 to obtain all solutions (x,y,n) of the equation for the fixed positive integers k≤10. In this paper, we improve the bound as n≤ 10,000 for the same case k≤10, and for any fixed general positive integer k, we give an upper bound depending only on k for n.

scholarly journals On the 2-adic order of Stirling numbers of the second kind and their differences

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Tamás Lengyel
Keyword(s):  
Positive Integer ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Binary Representation ◽  
Exact Order ◽  
Special Cases ◽  
International Audience ◽  
The Difference ◽  
Divisibility Properties ◽  
Positive Integers

International audience Let $n$ and $k$ be positive integers, $d(k)$ and $\nu_2(k)$ denote the number of ones in the binary representation of $k$ and the highest power of two dividing $k$, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that $\nu_2(S(2^n,k))=d(k)-1, 1\leq k \leq 2^n$. Here we prove that $\nu_2(S(c2^n,k))=d(k)-1, 1\leq k \leq 2^n$, for any positive integer $c$. We improve and extend this statement in some special cases. For the difference, we obtain lower bounds on $\nu_2(S(c2^{n+1}+u,k)-S(c2^n+u,k))$ for any nonnegative integer $u$, make a conjecture on the exact order and, for $u=0$, prove part of it when $k \leq 6$, or $k \geq 5$ and $d(k) \leq 2$. The proofs rely on congruential identities for power series and polynomials related to the Stirling numbers and Bell polynomials, and some divisibility properties.

scholarly journals Statistics on Lattice Walks and q-Lassalle Numbers

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Lenny Tevlin
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Integer Coefficients ◽  
Major Index ◽  
Lattice Walks ◽  
International Audience ◽  
The Difference ◽  
Positive Integers

International audience This paper contains two results. First, I propose a $q$-generalization of a certain sequence of positive integers, related to Catalan numbers, introduced by Zeilberger, see Lassalle (2010). These $q$-integers are palindromic polynomials in $q$ with positive integer coefficients. The positivity depends on the positivity of a certain difference of products of $q$-binomial coefficients.To this end, I introduce a new inversion/major statistics on lattice walks. The difference in $q$-binomial coefficients is then seen as a generating function of weighted walks that remain in the upper half-plan. Cet document contient deux résultats. Tout d’abord, je vous propose un $q$-generalization d’une certaine séquence de nombres entiers positifs, liés à nombres de Catalan, introduites par Zeilberger (Lassalle, 2010). Ces $q$-integers sont des polynômes palindromiques à $q$ à coefficients entiers positifs. La positivité dépend de la positivité d’une certaine différence de produits de $q$-coefficients binomial.Pour ce faire, je vous présente une nouvelle inversion/major index sur les chemins du réseau. La différence de $q$-binomial coefficients est alors considérée comme une fonction de génération de trajets pondérés qui restent dans le demi-plan supérieur.

scholarly journals On the Number of Witnesses in the Miller–Rabin Primality Test

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 890
Author(s):  
Shamil Talgatovich Ishmukhametov ◽  
Bulat Gazinurovich Mubarakov ◽  
Ramilya Gakilevna Rubtsova
Keyword(s):  
Positive Integer ◽  
Euler’S Totient Function ◽  
Number Of Factors ◽  
Primality Test ◽  
Average Probability ◽  
Test Errors ◽  
The Difference

In this paper, we investigate the popular Miller–Rabin primality test and study its effectiveness. The ability of the test to determine prime integers is based on the difference of the number of primality witnesses for composite and prime integers. Let W ( n ) denote the set of all primality witnesses for odd n. By Rabin’s theorem, if n is prime, then each positive integer a < n is a primality witness for n. For composite n, the power of W ( n ) is less than or equal to φ ( n ) / 4 where φ ( n ) is Euler’s Totient function. We derive new exact formulas for the power of W ( n ) depending on the number of factors of tested integers. In addition, we study the average probability of errors in the Miller–Rabin test and show that it decreases when the length of tested integers increases. This allows us to reduce estimations for the probability of the Miller–Rabin test errors and increase its efficiency.

A NOTE ON THE ERDŐS–GRAHAM THEOREM

2018 ◽  
Vol 97 (3) ◽  
pp. 363-366
Author(s):  
WENHUI WANG ◽  
MIN TANG
Keyword(s):  
Large Integer ◽  
Consecutive Integers ◽  
The Difference ◽  
Asymptotic Basis

Let ${\mathcal{A}}=\{a_{1}<a_{2}<\cdots \,\}$ be a set of nonnegative integers. Put $D({\mathcal{A}})=\gcd \{a_{k+1}-a_{k}:k=1,2,\ldots \}$. The set ${\mathcal{A}}$ is an asymptotic basis if there exists $h$ such that every sufficiently large integer is a sum of at most $h$ (not necessarily distinct) elements of ${\mathcal{A}}$. We prove that if the difference of consecutive integers of ${\mathcal{A}}$ is bounded, then ${\mathcal{A}}$ is an asymptotic basis if and only if there exists an integer $a\in {\mathcal{A}}$ such that $(a,D({\mathcal{A}}))=1$.

Complexe Woorden In Het Mentale Lexicon Van Kinderen

2006 ◽  
Vol 75 ◽  
pp. 53-65
Author(s):  
Maria Mos
Keyword(s):  
Family Size ◽  
Reading Ability ◽  
Reaction Times ◽  
Large Family ◽  
Decision Task ◽  
Large Family Size ◽  
Small Family Size ◽  
Good Reading ◽  
The Difference ◽  
Complex Words

Do children's representations of complex words in their mental lexicon have an internal structure, with the stem as a separate unit? De Jong et al (2000) found that adults recognize words with a large Family Size, i.e. words occurring in many derivations and compounds, faster than equally frequent words with a small Family Size. This result is an indication that the occurrence of a stem in complex words facilitates the recognition of this stem. This article investigates whether the Family-Size-effect extends to children's reaction times as well. Using a lexical decision task, the effect was observed in 9-10 year old children (N=69) in Dutch. A large vocabulary and good reading ability shortened general reaction times, but had no influence on the difference between items with a small or large Family Size. Monolingual and bilingual children performed similarly on this task.

On the k-th difference of an additive representation function

2011 ◽  
Vol 48 (1) ◽  
pp. 93-103
Author(s):  
Sándor Kiss
Keyword(s):  
Infinite Sequence ◽  
Probabilistic Methods ◽  
Number Of Solutions ◽  
Fixed Integer ◽  
Representation Function ◽  
Additive Representation ◽  
Consecutive Integers ◽  
Additive Representation Function ◽  
Positive Integers

Let k ≧ 2 be a fixed integer, A = {a1, a2, …} (a1 < a2 < …) be an infinite sequence of positive integers, and let Rk(n) denote the number of solutions of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$a_{i_1 } + a_{i_2 } + \cdots + a_{i_k } = n,a_{i_1 } \in \mathcal{A},...,a_{i_k } \in \mathcal{A}$$ \end{document}. Let B(A, N) denote the number of blocks formed by consecutive integers in A up to N. In [5], it was proved that if k > 2 and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\lim _{N \to \infty } \frac{{B(\mathcal{A},N)}}{{\sqrt[k]{N}}}$$ \end{document} = ∞ then |δl(Rk(n))| cannot be bounded for l ≦ k. The aim of this paper is to show that the above result is nearly best possible. We are using probabilistic methods.

Stability and Turán Numbers of a Class of Hypergraphs via Lagrangians

2017 ◽  
Vol 26 (3) ◽  
pp. 367-405 ◽  
Author(s):  
AXEL BRANDT ◽  
DAVID IRWIN ◽  
TAO JIANG
Keyword(s):  
Positive Integer ◽  
Structural Stability ◽  
Pairwise Disjoint ◽  
Large Family ◽  
Extremal Graphs ◽  
Uniform Hypergraphs ◽  
Turán Number

Given a family ofr-uniform hypergraphs${\cal F}$(orr-graphs for brevity), the Turán number ex(n,${\cal F})$of${\cal F}$is the maximum number of edges in anr-graph onnvertices that does not contain any member of${\cal F}$. A pair {u,v} iscoveredin a hypergraphGif some edge ofGcontains {u, v}. Given anr-graphFand a positive integerp⩾n(F), wheren(F) denotes the number of vertices inF, letHFpdenote ther-graph obtained as follows. Label the vertices ofFasv1,. . .,vn(F). Add new verticesvn(F)+1,. . .,vp. For each pair of verticesvi, vjnot covered inF, add a setBi,jofr− 2 new vertices and the edge {vi, vj} ∪Bi,j, where theBi,jare pairwise disjoint over all such pairs {i, j}. We callHFpthe expanded p-clique with an embedded F. For a relatively large family ofF, we show that for all sufficiently largen, ex(n,HFp) = |Tr(n, p− 1)|, whereTr(n, p− 1) is the balanced complete (p− 1)-partiter-graph onnvertices. We also establish structural stability of near-extremal graphs. Our results generalize or strengthen several earlier results and provide a class of hypergraphs for which the Turán number is exactly determined (for largen).

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