Positive Integer Research Articles

A further generalization of sums of higher derivatives of the Riemann zeta function

We obtain a full asymptotic for the sum of [Formula: see text], where [Formula: see text] denotes the [Formula: see text]th derivative of the Riemann zeta function, [Formula: see text] is a positive real number, and [Formula: see text] denotes a nontrivial zero of the Riemann zeta function. The sum is over the zeros with imaginary parts up to a height [Formula: see text], as [Formula: see text]. We also specify what the asymptotic formula becomes when [Formula: see text] is a positive integer, highlighting the differences in the asymptotic expansions as [Formula: see text] changes its arithmetic nature.

Proof of a conjecture on congruences of the sum of minimal excludants

Very recently, Baruah et al. refined a result of Andrews and Newman on the sum of minimal excludants over all the partitions of a positive integer [Formula: see text] by establishing the generating functions of two functions [Formula: see text] and [Formula: see text] which denote the sum of odd minimal excludants and the sum of even minimal excludants, respectively. They proved some congruences on [Formula: see text] and [Formula: see text] modulo 4 and 8 and posed a conjecture on [Formula: see text] and [Formula: see text] at the end of their paper. In this paper, we confirm this conjecture by using theta function identities.

Log-Concavity with Respect to the Number of Orbits for Infinite Tuples of Commuting Permutations

AbstractLet A(p, n, k) be the number of p-tuples of commuting permutations of n elements whose permutation action results in exactly k orbits or connected components. We formulate the conjecture that, for every fixed p and n, the A(p, n, k) form a log-concave sequence with respect to k. For $$p=1$$ p = 1 this is a well-known property of unsigned Stirling numbers of the first kind. As the $$p=2$$ p = 2 case, our conjecture includes a previous one by Heim and Neuhauser, which strengthens a unimodality conjecture for the Nekrasov–Okounkov hook length polynomials. In this article, we prove the $$p=\infty $$ p = ∞ case of our conjecture. We start from an expression for the A(p, n, k), which follows from an identity by Bryan and Fulman, obtained in the their study of orbifold higher equivariant Euler characteristics. We then derive the $$p\rightarrow \infty $$ p → ∞ asymptotics. The last step essentially amounts to the log-concavity in k of a generalized Turán number, namely, the maximum product of k positive integers whose sum is n.

On a Problem of Douglass and Ono for the Plane Partition Function

AbstractIt is known that the plane partition function of n denoted $$\textrm{PL}(n)$$ PL ( n ) obeys Benford’s law in any integer base $$b\ge 2$$ b ≥ 2 . We give an upper bound for the smallest positive integer n such that $$\textrm{PL}(n)$$ PL ( n ) starts with a prescribed string f of digits in base b.

The Bounds of Rigid Sphere Design

AbstractWe give some necessary conditions for the existence of rigid sphere designs. In particular, we settle the conjecture proposed by Bannai (J Fac Sci Univ Tokyo Sect IA Math 34(3), 485-489, 1987). Given fixed positive integers t and d, we show that there exist only finitely many rigid t-designs on $$S^d$$ S d , up to orthogonal transformations.

Collatz Conjecture

This paper presents an analysis of the number of zeros in the binary representation of natural numbers. The primary method of analysis involves the use of the concept of the fractional part of a number, which naturally emerges in the determination of binary representation. This idea is grounded in the fundamental property of the Riemann zeta function, constructed using the fractional part of a number. Understanding that the ratio between the fractional and integer parts of a number, analogous to the Riemann zeta function, reflects the profound laws of numbers becomes the key insight of this work. The findings suggest a new perspective on the interrelation between elementary properties of numbers and more complex mathematical concepts, potentially opening new directions in number theory and analysis.This analysis has allowed to understand that the Collatz sequence initially tends towards a balanced symmetric arrangement of zeros and ones, and then it collapses, realizing the scenario of the Collatz conjecture.

Algorithms for representing positive odd integers as the sum of arithmetic progressions

This paper delves into the historical and recent developments in this area of mathematical inquiry, tracing the evolution from Wheatstone’s representation of powers of an integer as sums of arithmetic progressions to extensions of Sylvester’s Theorem (Sylvester and Franklin, [14]). Sylvester’s Theorem, a result that determines the representability of positive integers as sums of consecutive integers, has been the foundation for numerous extensions, including the representation of integers as sums of specific arithmetic progressions and powers of such progressions. The recent works of Ho et al. [3] and Ho et al. [4] have further expanded on Sylvester’s Theorem, offering a procedural approach to compute the representability of positive integers in the context of arithmetic progressions. In this paper, efficient algorithms to compute the number of ways to represent an odd positive integer as sums of powers of arithmetic progressions are presented.

On a union of homogeneous sequences

A sequence of positive integers is said to be quasi-primitive if there do not exist three distinct integers such that the greatest common divisor of two elements is not the third. It is clear that the sequence of prime numbers is quasi-primitive. The Erdős quasi-primitive conjecture states that the sum of the reciprocal of the product of an element by its logarithm over any quasi-primitive sequence is maximized by the sum over the sequence of all powers of prime numbers. Note that Erdős, before introduce his conjecture, showed that for any quasi-primitive sequence different from one, this sum is convergent and bounded by a real constant. In this article, by studying the analogous Erdős conjecture, we prove that when the real is sufficiently large, there exists a quasi-primitive sequence such that the sum of the reciprocal of the product of an element by the sum of its logarithm and , over this quasi-primitive sequence is very large than the sum over the sequence of all powers of prime numbers. Obviously, if the case where equals zero is also true, then the conjecture is false.

On singular foliations tangent to a given hypersurface

We consider a class of singular foliations in the sense of Androulidakis and Skandalis that we call transverse order k foliations. These have a finite number of leaves: one hypersurface (the singular leaf) together with the components of its complement (open leaves). The positive integer parameter k encodes the “order of tangency” of the leafwise vector fields to L . We show that a loop in the singular leaf induces a well-defined holonomy transformation at the level of (k-1) -jets. The resulting holonomy invariant can be used to give a complete classification of these foliations and obtain concrete descriptions of their associated groupoids and algebras.

Frobenius Local Rings of Order p4m

Suppose R is a finite commutative local ring, then it is known that R has four positive integers p,n,m,k called the invariants of R, where p is a prime number. This paper investigates the structure and classification up to isomorphism of local rings with residue field Fpm and of length 4. Specifically, it gives a comprehensive characterization of Frobenius local rings of order p4m. Furthermore, we provide a detailed enumeration of the classes of all such rings with respect to their invariants p,n,m,k. Finite Frobenius rings are particularly advantageous for coding theory. This suitability arises from the fact that two classical theorems by MacWilliams, the Extension Theorem and the MacWilliams relations for symmetrized weight enumerators, can be generalized from finite fields to finite Frobenius rings.

On monogenity of certain number fields defined by x 2·3 s ·q r − m

Let K = Q ( α ) , where α is a root of the monic irreducible polynomial x 2 · 3 s · q r − m with m ≠ ± 1 is a square-free integer, r and s are two positive integers, and q is a prime of the form 3 k + 2 . In this article, we study the monogenity of the number field K and establish conditions on m for K to be monogenic. Further, we provide sufficient conditions on r, s and m for K to be not monogenic. Finally, we illustrate our results with examples.

On the periodic behavior of the generalized Chazy differential equation.

We consider the periodic behavior of the generalized Chazy differential equation x⃛+|x|qx¨+k|x|qxx˙2=0, where q is a positive integer and k is a real number. We give a pure analysis on the existence of non-trivial periodic solutions for k=q+1 and the non-existence of them for k≠q+1. Our method is based on considering the projections of the orbits onto the phase plane (x,x˙). We find that a non-trivial periodic solution of the equation is equivalent to a closed curve formed by two equilibrium points and two orbits with some specific constraints in the corresponding planar system and that the existence of such closed curves can be obtained by the existence of real zeros of some returning map. Our conclusion covers all q, which completes a recent result.

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.

On the existence of certain Lehmer numbers modulo a prime

A Lehmer number modulo an odd prime numberp is a residue class a∈Fp× whose multiplicative inverse ā has opposite parity. Lehmer numbers that are also primitive roots are called Lehmer primitive roots. Analogously, in this article we say that a residue class a∈Fp× is a Lehmer non-primitive root modulop if a is Lehmer number modulo p which is not a primitive root. We provide explicit estimates for the difference between the number of Lehmer non-primitive roots modulo a prime p and their “expected number”, which is p−1−ϕ(p−1)2. Similar explicit bounds are also provided for the number of k-consecutive Lehmer numbers modulo a prime, and k-consecutive Lehmer primitive roots We also prove that for any prime number p>3.05×1014, there exists a Lehmer non-primitive root modulo p. Moreover, we show that for any positive integer k≥2 (respectively, k≥5) and for all primes p≥exp(122k3) (respectively, p≥exp(6.87k)), there exist k consecutive Lehmer numbers modulo p (respectively, k consecutive Lehmer primitive roots modulo p). For large primes p, these theorems generalize two results which were proven in a paper by Cohen and Trudgian appeared in the Journal of Number Theory in 2019.

Solution of Well-known Physical Problems with Fractional Conformable Derivative

This paper presents a new definition of fractional derivatives and integrals through the conformable derivative approach. This innovative framework offers a closer alignment with classical derivative concepts while providing a more practical and intuitive basis for fractional calculus. The new definition is applicable in two primary ranges: 0 ≤ α < 1 and n − 1 ≤ α < n, where n is a positive integer. It is shown that when α = 1, this definition corresponds precisely to the classical first-order derivative. Key benefits of this approach include its improved consistency with traditional calculus and greater computational ease, making it a useful tool for both theoretical research and practical applications. By integrating fractional calculus with conventional derivative ideas, this definition simplifies the analysis and interpretation of fractional differential equations and their solutions. Additionally, we examine the definition’s effects on stability and convergence in numerical methods and provide examples demonstrating its effectiveness and applicability.

Degenerate Moments and Expectation of Monomials

The aim of this paper is twofold. Firstly, we obtain expressions of the degenerate moments of a discrete nonnegative integer valued random variable. Secondly, we get an expression for the expectation of any monomial in discrete nonnegative integer-valued random variables.

Some Generator Subgraphs of the Square of a Cycle

Graphs considered in this paper are finite simple graphs, which has no loops and multiple edges. Let $G= (V(G),E(G))$ be a graph with $E(G) = \{e_{1}, e_{2},\ldots, e_{m}\},$ for some positive integer $m.$ The \textit{edge space} of $G,$ denoted by $\mathscr{E}(G),$ is a vector space over the field $\zn_{2}.$ The elements of $\mathscr{E}(G)$ are all the subsets of $E(G)$. Vector addition is defined as $X+Y = X ~\Delta~ Y,$ the symmetric difference of sets $X$ and $Y,$ for $X,Y \in \mathscr{E}(G).$ Scalar multiplication is defined as $1\cdot X =X$ and $0 \cdot X = \emptyset$ for $X \in \mathscr{E}(G).$ Let $H$ be a subgraph of $G.$ The \textit{uniform set of $H$} with respect to $G,$ denoted by $E_{H}(G),$ is the set of all elements of $\mathscr E(G)$ that induces a subgraph isomorphic to $H.$ The subspace of $\mathscr E(G)$ generated by $E_{H}(G)$ shall be denoted by $\mathscr E_{H}(G).$ If $E_H(G)$ is a generating set, that is $\mathscr E_{H}(G)= \mathscr E(G),$ then $H$ is called a \textit{generator subgraph} of $G.$ This paper determines some generator subgraphs of the square of a cycle. Moreover, this paper established sufficient conditions for the generator subgraphs of the square of a cycle.

Contractibility of the Rips complexes of Integer lattices via local domination

We prove that for each positive integer n n , the Rips complexes of the n n -dimensional integer lattice in the d 1 d_1 metric (i.e., the Manhattan metric, also called the natural word metric in the Cayley graph) are contractible at scales above n 2 ( 2 n − 1 ) n^2(2n-1) , with the bounds arising from the Jung constants. We introduce a new concept of locally dominated vertices in a simplicial complex, upon which our proof strategy is based. This allows us to deduce the contractibility of the Rips complexes from a local geometric condition called local crushing. In the case of the integer lattices in dimension n n and a fixed scale r r , this condition entails the comparison of finitely many distances to conclude that the corresponding Rips complex is contractible. In particular, we are able to verify that for n = 1 , 2 , 3 n=1,2,3 , the Rips complex of the n n -dimensional integer lattice at scale greater or equal to n n is contractible. We conjecture that the same proof strategy can be used to extend this result to all dimensions n n .

Statistical Analysis and Distribution of Fermat Pseudoprimes Within the Given Interval

Prime numbers are natural numbers that can only be divided by one and the original number. There is more than one of them. Error-correcting codes used in telecommunications are generated using prime numbers. They guarantee automatic message correction both during transmission and reception. Algorithms used in public-key cryptography are built upon primes. They're also employed in the production of pseudorandom numbers. Mathematicians and many other scientific and technological communities have always been fascinated by prime numbers. Additionally, computer engineers can use it to tackle a wide range of real-world problems. The analysis of prime numbers is very important for finding their applications in different fields. The statistical analysis of pseudoprimes within a given interval is carried out in the presented paper and an algorithm of Python program to find the distribution of pseudoprimes has also been generated, which is used to find their distribution with different bases within the given intervals. The data analysis process made use of graphical depiction. The discovery will surely open up new avenues for future number theory study and applications outside of mathematics.

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.