Positive Integer Research Articles

The average number of integral points on the congruent number curves

We show that the total number of non-torsion integral points on the elliptic curves ED:y2=x3−D2x, where D ranges over positive squarefree integers less than N, is O(N(log⁡N)−14+ϵ). The proof involves a discriminant-lowering procedure on integral binary quartic forms and an application of Heath-Brown's method on estimating the average size of the 2-Selmer groups of the curves in this family.

An Elementary Proof for Fermat's Last Theorem using Ramanujan-Nagell Equation

Fermat’s Last Theorem states that it is impossible to find positive integers A, B and C satisfying the equation An + Bn = Cn where n is any integer > 2. Taking the proofs of Fermat for the index n = 4, and Euler for n = 3, it is sufficient to prove the theorem for n = p, any prime > 3. We hypothesize that all r, s and t are non-zero integers in the equation rp + sp = tp and establish a contradiction in this proof. Just for supporting the proof in the above equation, we have used another equation x3 + y3 = z3 Without loss of generality, we assert that both x and y as non-zero integers; z3 a non-zero integer; z and z2 irrational. We create transformed equations to the above two equations through parameters, into which we have incorporated the Ramanujan - Nagell equation. Solving the transformed equations we prove the theorem.

Degrees of relations on canonically ordered natural numbers and integers

AbstractWe investigate the degree spectra of computable relations on canonically ordered natural numbers $$(\omega ,<)$$ ( ω , < ) and integers $$(\zeta ,<)$$ ( ζ , < ) . As for $$(\omega ,<)$$ ( ω , < ) , we provide several criteria that fix the degree spectrum of a computable relation to all c.e. or to all $$\Delta _2$$ Δ 2 degrees; this includes the complete characterization of the degree spectra of so-called computable block functions that have only finitely many types of blocks. Compared to Bazhenov et al. (in: LIPIcs, vol 219, pp 8:1–8:20, 2022), we obtain a more general solution to the problem regarding possible degree spectra on $$(\omega ,<)$$ ( ω , < ) , answering the question whether there are infinitely many such spectra. As for $$(\zeta ,<)$$ ( ζ , < ) , we prove the following dichotomy result: given an arbitrary computable relation R on $$(\zeta ,<)$$ ( ζ , < ) , its degree spectrum is either trivial or it contains all c.e. degrees. This result, and the proof techniques required to solve it, extend the analogous theorem for $$(\omega ,<)$$ ( ω , < ) obtained by Wright (Computability 7:349–365, 2018), and provide initial insight to Wright’s question whether such a dichotomy holds on computable ill-founded linear orders. This article is an extended version of Bazhenov et al. (in: LIPIcs, vol 219, pp 8:1–8:20, 2022).

Zero-sum partitions of Abelian groups and their applications to magic- and antimagic-type labelings

The following problem has been known since the 80s. Let $\Gamma$ be an Abelian group of order $m$ (denoted $|\Gamma|=m$), and let $t$ and $\{m_i\}_{i=1}^{t}$, be positive integers such that $\sum_{i=1}^t m_i=m-1$. Determine when $\Gamma^*=\Gamma\setminus\{0\}$, the set of non-zero elements of $\Gamma$, can be partitioned into disjoint subsets $\{S_i\}_{i=1}^{t}$ such that $|S_i|=m_i$ and $\sum_{s\in S_i}s=0$ for every $1 \leq i \leq t$. Such a subset partition is called a \textit{zero-sum partition}. $|I(\Gamma)|\neq 1$, where $I(\Gamma)$ is the set of involutions in $\Gamma$, is a necessary condition for the existence of zero-sum partitions. In this paper, we show that the additional condition of $m_i\geq 4$ for every $1 \leq i \leq t$, is sufficient. Moreover, we present some applications of zero-sum partitions to magic- and antimagic-type labelings of graphs.

Most integers are not a sum of two palindromes

Abstract For $g \geqslant 2$ , we show that the number of positive integers at most X which can be written as sum of two base g palindromes is at most ${X}/{\log^c X}$ . This answers a question of Baxter, Cilleruelo and Luca.

Sidon–Ramsey and $$B_{h}$$-Ramsey numbers

AbstractFor a given positive integer k, the Sidon–Ramsey number $${{\,\textrm{SR}\,}}(k)$$ SR ( k ) is defined as the minimum value of n such that, in every partition of the set [1, n] into k parts, there exists a part that contains two distinct pairs of numbers with the same sum, i.e., one of the parts is not a Sidon set. In this paper, we investigate the asymptotic behavior of this parameter and two generalizations of it. The first generalization involves replacing pairs of numbers with h-tuples, such that in every partition of [1, n] into k parts, there exists a part that contains two distinct h-tuples with the same sum, i.e., there is a part that is not a $$B_h$$ B h set. The second generalization considers the scenario where the interval [1, n] is substituted with a d-dimensional box of the form $$\prod _{i=1}^d[1,n_i]$$ ∏ i = 1 d [ 1 , n i ] . For the general case of $$h\ge 3$$ h ≥ 3 and d-dimensional boxes, before applying our method to obtain the Ramsey-type result, we establish an upper bound for the corresponding density parameter.

Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions

For positive integers $d$ and $p$ such that $d \ge p$, let $\mathbb{R}^{d \times p}$ denote the set of $d \times p$ real matrices, $I_p$ be the identity matrix of order $p$, and $V_{d,p} = \{x \in \mathbb{R}^{d \times p} \mid x'x = I_p\}$ be the Stiefel manifold in $\mathbb{R}^{d \times p}$. Complete asymptotic expansions as $d \to \infty$ are obtained for the normalizing constants of the matrix Bingham and matrix Langevin probability distributions on $V_{d,p}$. The accuracy of each truncated expansion is strictly increasing in $d$; also, for sufficiently large $d$, the accuracy is strictly increasing in $m$, the number of terms in the truncated expansion. Lower bounds are obtained for the truncated expansions when the matrix parameters of the matrix Bingham distribution are positive definite and when the matrix parameter of the matrix Langevin distribution is of full rank. These results are applied to obtain the rates of convergence of the asymptotic expansions as both $d \to \infty$ and $p \to \infty$. Values of $d$ and $p$ arising in numerous data sets are used to illustrate the rate of convergence of the truncated approximations as $d$ or $m$ increases. These results extend recently-obtained asymptotic expansions for the normalizing constants of the high-dimensional Bingham distributions.

On the Within-Perfect Numbers

Abstract Motivated by the works of Erdös, Wirsing, Pomerance, Wolke and Harman on the sum-of-divisor function $\sigma(n)$, we study the distribution of a special class of natural numbers closely related to (multiply) perfect numbers which we term ‘$(\ell;k)$-within-perfect numbers’, where $\ell \gt 1$ is a real number and $k: [1, \infty) \rightarrow (0, \infty)$ is an increasing and unbounded function.

Multiplication operators on the weighted Sobolev disk algebra

Let ⅅ be the unit disk in the complex plane ℂ. For α > −1, the weighted Sobolev disk algebra SA(ⅅ, dAα ) consists of all analytic functions in the weighted Sobolev space W 2,2(ⅅ, dAα ). In this paper, we prove that the multiplication operator is similar to on SA(ⅅ, dAα ) if and only if n = m, where n, m are positive integers. Then we characterize when a bounded operator P on SA(ⅅ, dAα ) belongs to the commutant of by using the matrix representation of P. In addition, we compute the exact norm of Mz on SA(ⅅ, dAα ). Finally, we prove that on the unweighted Sobolev disk algebra SA(ⅅ) the restrictions of to different invariant subspaces zkSA(ⅅ) (k ≥ 1) are not unitarily equivalent, and the restrictions of (n ≥ 2) to different invariant subspaces Sj (0 ≤ j < n) are also not unitarily equivalent.

Periodic derived Hall algebras of hereditary abelian categories

Let m be a positive integer and Dm(A) be the m-periodic derived category of a finitary hereditary abelian category A. Applying the derived Hall numbers of the bounded derived category Db(A), we define an m-periodic extended derived Hall algebra for Dm(A), and use it to give a global, unified and explicit characterization for the algebra structure of Bridgeland's Hall algebra of periodic complexes. Moreover, we also provide an explicit characterization for the odd periodic derived Hall algebra of A defined by Xu-Chen [24].

Let ‘Let n be such an x’ Be: A Reply to Meléndez Gutiérrez

AbstractI defend Haze’s argument against the Breckenridge-Magidor theory of instantial reasoning from an objection by Meléndez Gutiérrez. According to Breckenridge and Magidor, in reasoning like ‘Some x is mortal. Let n be such an x…’, the ‘n’ refers to a particular object but we cannot know which. This surprisingly defensible view poses an obvious threat to widespread notions in the philosophy of language. Haze argues that the theory leads to absurdity in cases like ‘Some x is unreferred-to by any expression. Let n be such an x…’ and should therefore be rejected. Meléndez Gutiérrez counters that Haze’s argument is just a case of Berry-like paradox and thus fails to refute the Breckenridge-Magidor theory. I argue that the analogy breaks down: unlike the intuitively compelling and widely believed well-ordering principle about positive integers, the principle drawn from Breckenridge and Magidor that plays a supposedly analogous role enjoys no such status, and is instead simply shown to be false by Haze’s reductio. The possibility of such a response is obscured when Meléndez Gutiérrez portrays Haze’s argument as involving a stipulation that ‘n’ is to refer to an unreferred-to object. On the contrary, Haze’s argument does not assume that expressions like ‘n’ work by means of referring at all, and simply lets stipulations like ‘Let n be such an x’ be themselves, without imposing a theory on them. Once this is clarified, we can see that Haze’s argument is unaffected by Meléndez Gutiérrez’s objection.

Powers as Fibonacci sums

We examine the equation for positive integers y, a ≥ 2 and k ≥ 3. This equation can be expressed as a problem in terms of the Zeckendorf representations of integers. Using bounds on linear forms in logarithms and Baker-Davenport reduction methods, we are able to completely solve the equation for y ≤ 25000 when k = 3, for y ≤ 1000 when k = 4, for y ≤ 40 when k = 5, and for y ≤ 3 when k = 6.

EDGE IRREGULAR REFLEXIVE LABELING OF DUMBBELL GRAPH, CORONA OF OPEN LADDER, AND NULL GRAPH

Graph is a simple, connected, undirected graph with vertex set and edge set . A graph is called to have an edge irregular reflexive -labeling if its vertices can be labeled with even numbers from until and its edges can be labeled with positive integers from to such that the weights for all the edges are different, where . The weight of edge uv in graph with labeling, denoted by , is defined as sum of the edge label and all vertex labels incident to that edge. The reflexive edge strength of a graph , denoted by , is the value of minimum of the largest label. In this paper, edge irregular reflexive -labeling for Dumbbell Graph and corona of open ladder and null graph will be determined. The reflexive edge strength of the Dumbbell Graph with and is for and for The reflexive edge strength of the corona of open ladder and null graph with n ≥ 3 and m ≥ 1 is for and for .

THE REFLEXIVE EDGE STRENGTH OF THE PENTAGONAL SNAKE GRAPH AND CORONA OF THE OPEN TRIANGULAR LADDER AND NULL GRAPH

Assume that be an undirected simple graph with vertex set and edge set . The edge irregular reflexive -labeling of graph is a labeling selects positive integers from 1 to as edge labels and non negative even numbers from 0 to as vertex labels, and the weights assigned to each edge are distinct, where . On graph with labeling, the weight of edge is represented by which is defined as the sum of edge label and all vertex labels incident to that edge. Reflexive edge strength of graph is the minimum of the highest label, denoted by . In this research, reflexive edge strength for pentagonal snake graph and corona of open triangular ladder and null graph will be determined. The method of this research is literature study, the lower bound of determined by Ryan’s lemma and the upper bound by labeling. The reflexive edge strength of pentagonal snake graph with is for and for The reflexive edge strength of corona of open triangular ladder and null graph with n ≥ 3 and m ≥ 1 is and .

On the Conflation of Negative Binomial and Logarithmic Distributions

In recent decades, the study of discrete distributions has received increasing attention in the field of statistics, mainly because discrete distributions can model a wide range of count data. One common distribution used for modeling count data, for instance, is the negative binomial distribution (NBD), which performs well with over-dispersed data. In this paper, a new count distribution is introduced, called the conflation of negative binomial and logarithmic distributions, which is formed by conflating the negative binomial and logarithmic distributions, resulting in a distribution that possesses some of the properties of negative binomial and logarithmic distributions. The distribution has two parameters and is verified by a positive integer. Two modifications are proposed to the distribution, which includes zero as a support point. The new distribution is valuable from a theoretical perspective since it is a member of the weighted negative binomial distribution family. In addition, the distribution differs from the NBD in the sense that the probability of lower counts is inflated. This study discusses the characteristics of the proposed distribution and its modified versions, such as moments, probability generating functions, likelihood stochastic ordering, log-concavity, and unimodality properties. Real-world data are used to evaluate the performance of the proposed models against other models. All computations shown in this paper were produced using the R programming language.

On the Balance between Emigration and Immigration as Random Walks on Non-Negative Integers

Life is on a razor’s edge resulting from the random competitive forces of birth and death. We illustrate this aphorism in the context of three Markov chain population models where systematic random immigration events promoting growth are simultaneously balanced with random emigration ones provoking thinning. The origin of mass removals is either determined by external demands or by aging, leading to different conditions of stability.

Energy Constrained Depth First Search

AbstractDepth first search is a natural algorithmic technique for constructing a closed route that visits all vertices of a graph. The length of such a route equals, in an edge-weighted tree, twice the total weight of all edges of the tree and this is asymptotically optimal over all exploration strategies. This paper considers a variant of such search strategies where the length of each route is bounded by a positive integer B (e.g. due to limited energy resources of the searcher). The objective is to cover all the edges of a tree T using the minimum number of routes, each starting and ending at the root and each being of length at most B. To this end, we analyze the following natural greedy tree traversal process that is based on decomposing a depth first search traversal into a sequence of limited length routes. Given any arbitrary depth first search traversal R of the tree T, we cover R with routes $$R_1,\ldots ,R_l$$ R 1 , … , R l , each of length at most B such that: $$R_i$$ R i starts at the root, reaches directly the farthest point of R visited by $$R_{i-1}$$ R i - 1 , then $$R_i$$ R i continues along the path R as far as possible, and finally $$R_i$$ R i returns to the root. We call the above algorithm piecemeal-DFS and we prove that it achieves the asymptotically minimal number of routes l, regardless of the choice of R. Our analysis also shows that the total length of the traversal (and thus the traversal time) of piecemeal-DFS is asymptotically minimum over all energy-constrained exploration strategies. The fact that R can be chosen arbitrarily means that the exploration strategy can be constructed in an online fashion when the input tree T is not known in advance. Each route $$R_i$$ R i can be constructed without any knowledge of the yet unvisited part of T. Surprisingly, our results show that depth first search is efficient for energy constrained exploration of trees, even though it is known that the same does not hold for energy constrained exploration of arbitrary graphs.

Random balancing-like sequences

AbstractA balancing-like sequence is a binary recurrence sequence which generalizes the balancing sequence and the sequence of nonnegative integers. This sequence, under certain assumptions, may be used to describe the growth of fortune of a person engaged in some business or profession. Since, in any business or profession, the growth is influenced by many uncertainties, it is more natural to induce some sort of randomness in the balancing-like sequences. If, in a balancing-like sequence, the growth rate is assumed to be a random variable, the resulting sequence will be a stochastic process and the sequence of expectations, in many cases, cannot be described as a binary recurrence sequence. In some cases, the growth rate of expectations increases without limit while, in some cases, it remains finite.

Vertex-critical graphs far from edge-criticality

Abstract Let $r$ be any positive integer. We prove that for every sufficiently large $k$ there exists a $k$ -chromatic vertex-critical graph $G$ such that $\chi (G-R)=k$ for every set $R \subseteq E(G)$ with $|R|\le r$ . This partially solves a problem posed by Erdős in 1985, who asked whether the above statement holds for $k \ge 4$ .

Partial actions of groups on generalized matrix rings

Let n be a positive integer and R=(Mij)1≤i,j≤n be a generalized matrix ring. For each 1≤i,j≤n, let Ii be an ideal of the ring Ri:=Mii and denote Iij=IiMij+MijIj. We give sufficient conditions for the subset I=(Iij)1≤i,j≤n of R to be an ideal of R. Also, suppose that α(i) is a partial action of a group G on Ri, for all 1≤i≤n. We construct, under certain conditions, a partial action γ of G on R such that γ restricted to Ri coincides with α(i). We study the relation between this construction and the notion of Morita equivalent partial group action given in [1]. Moreover, we investigate properties related to Galois theory for the extension Rγ⊂R. Some examples to illustrate the results are considered in the last part of the paper.