Aug 2026· Earthline Journal of Mathematical Sciences· pp. 1009· 0 citations· 15 references
Abstract
The study of integer $I$ for which $I$ is a sum of cubes is a subject of study in the areas of Diophantine equations, perhaps this is due to the fact that Diophantine equations have direct applications in coding theory. Let $r, w, d$ and $m$ be positive integers and suppose that $d$ is a common difference between any two integers in the sequence $w_{i+1}$ and $w_i$, where $m=w_{r+1}$ denotes the middle term of the sequence. The current paper is therefore set to develop and introduce the general formula for the sum of an odd number of centred odd cubes. In particular, the formula $\sum_{i=1}^{2r+1}w_i^{3}=\left(\sum_{i=1}^{2r+1}w_i\right)\left(m^{2}+r(r+1)d^{2}\right)$ is introduced. The methodology involves expressing the arithmetic sequence in its standard form, expanding the cube terms, and matching coefficients. Several examples are presented to verify the identity and illustrate its computational efficiency.
Let $M(N)$ be the number of similarity classes of positive rational perfect cuboids with ordered edges $a,b,c$, space diagonal $g$, and squarefree invariant $N=\operatorname{sf}(abcg/2)$. We prove that $M(N)$ is finite for every positive squarefree $N$, without any restriction on the rank of the associated congruent-nu...
Let $N$ be a sufficiently large real number. In this paper, it is proved that, for $1<c<\frac{5179}{3909}$ and for any arbitrary large number $E>0$, the Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p_3^c+p_4^c-N\big|<(\log N)^{-E} \end{equation*} is solvable in prime variables $p_1,p_2,p_3,p_4$ such that e...
Yu-Hui Zhao, Jin-Jia Li, Li Long et al.· 0 citations
In this paper, we study two questions concerning perfect powers in $S$-unit equations and sparse representations of perfect powers. Corvaja-Zannier \cite{corvaja2013finiteness} proved that there are only finitely many odd perfect powers in $\N$ having precisely four non-zero digits in their binary expansion. At first,...
N. Darsana, Parvathi S. Nair, S. Rout· 0 citations
Let $d(n)$ be the divisor function and denote by $[t]$ the integral part of the real number $t$. We prove that \begin{align*} \sum_{n\leq x^{1/c}}d\left(\left[\frac{x}{n^c}\right]\right)=d_cx^{1/c}+O_{\varepsilon,c}(x^{\theta_c+\varepsilon}), \end{align*} where $d_c$ is a suitable constant, $\theta_c<2/(3c+2)$ for $0<c...
Consider the polynomial $f=x^N+y^N-z^N$ where $x,\,y$ and $z$ are positive integers and $N \ge 3$ is an integer. By Fermat's Last Theorem, $f$ is never zero so that its reciprocal, $1/f$, has no singularities. We therefore study the finite sum of the reciprocal: $S(m,N)=\sum_{x=1}^m\sum_{y=1}^m\sum_{z=1}^{m}\frac{1}{f}...
Let $\nu_2(m)$ denote the $2$-adic valuation of a positive integer $m$ and set $N_m=m\bigl(1+\nu_2(m)/2\bigr)$. We prove four conjectures of Merca on the nonnegativity of truncated pentagonal number series weighted by the infinite products $\prod_{m\ge 1}(1-q^{2N_m})$ and $\prod_{r\ge 1}\bigl(q^{2^r r};q^{2^{r+1}r}\big...
Manjil P. Saikia, Abhishek Sarma· 0 citations
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