Earthline Journal of Mathematical Sciences

A Note on a Lebesgue-Ramanujan-Nagell Type Diophantine Equation

2026-09-07T15:33:17+00:00

We determine all solutions of
\[
x^2+3^a7^b37^c=\lambda y^n,
\]
where $\lambda\in\{1,2,4\}$, $x,y\geq1$, $a,b,c\geq0$, $n\geq3$, and $\gcd(x,y)=1$, subject to the following parity condition: when $\lambda\in\{1,2\}$ and neither $3$ nor $4$ divides $n$, the integer $y$ is assumed to be odd. The cases divisible by $3$ or $4$ are reduced to the determination of $S$-integral points on elliptic and quartic curves, with $S=\{3,7,37\}$. For the remaining exponents, a reduction to odd prime exponents is combined with the primitive divisor theorem for Lehmer sequences and the corrected classification of defective Lehmer pairs. All computationally obtained solutions are then checked directly in the original equation.

On an Inequality Analogous to the Nesbitt Inequality

2026-09-07T16:16:40+00:00

In this paper, we first examine a cyclic inequality analogous to the Nesbitt inequality, and subsequently establish generalized versions using convex functions and the Jensen inequality. Several applications are provided to illustrate the main results.

Linear-Interpolative Contraction Mapping Theorem for the Berinde Weak, Kannan, Ciric-Reich-Rus, and Chatterjea Operators in Metric Spaces with Application

2026-09-07T16:43:04+00:00

Motivated by the definition of interpolative metric space [6], in which the triangle inequality has been modified into an inequality consisting of linear and interpolative terms, we introduce in this paper the notion of linear-interpolative contractions which is analogous to the triangle inequality in the definition of interpolative metric space. These inequalities consist of linear and interpolative terms. Some results related to the linear-interpolative contractions are obtained in the setting of metric spaces with an illustrative example. Finally, we apply our result to the Fredholm integral equation.

Connectivity and Distance Properties of Zero-Divisor Graphs of Finite Commutative Semilocal Rings

2026-09-13T16:51:24+00:00

The zero-divisor graph of a commutative ring provides a natural connection between algebraic and graph-theoretic structures. Although extensive research has been conducted on the algebraic and combinatorial properties of zero-divisor graphs, their connectivity and metric properties over finite semilocal rings remain comparatively less explored. In this paper, we investigate the zero-divisor graph associated with the finite semilocal ring

R=\mathbb Z_{p^{n}q^{m}},

where $p$ and $q$ are distinct primes and $n,m\ge2$. Using a valuation-theoretic approach, we establish a natural decomposition of the vertex set into valuation layers. This decomposition enables us to determine the connectivity, shortest-path structure, eccentricities of valuation layers, diameter, radius, centre, and peripheral vertices of the graph. The results demonstrate that the valuation-layer decomposition completely governs the structural, metric, and symmetry properties of the zero-divisor graph, providing a unified framework for the study of finite semilocal rings and extending several known results on zero-divisor graphs.

Analytical Solutions of Volterra Integral Equations Involving Jacobi Polynomial Nonlinearities: A Power Series Method

2026-09-14T02:44:59+00:00

Nonlinear Volterra integral equations model several phenomena in a variety of fields, including epidemiology, population growth theory, diffusion and heat transfer problems. In this paper, the analytical solution of a class of nonlinear Volterra integral equations of the second kind is presented using a power series method based on the generalised Cauchy product. The nonlinear terms $\Phi(U(\xi))$ in the proposed problem are given by normalised Jacobi polynomials $\mathsf{P}_{\ell}^{(\phi, \varphi)}(U(\xi))$ ($\ell=1,2,3,\dots;\phi,\varphi>-1$) in the unknown $U(\xi)$. The proposed method first expresses the nonlinear terms as a power series in the unknown $U$, and then uses the generalised Cauchy product to convert the series into power series in the independent variable $\xi$. Subsequently, recurrence relations for the expansion coefficients of the series solution are obtained in terms of the nonlinear terms and their higher derivatives evaluated at the constant term of the series solution. Specialising the nonlinear terms to Gegenbauer or ultraspherical polynomials, Legendre polynomials, and Chebyshev polynomials, several nonlinear Volterra integral equations and their solutions are introduced. Considering the normalised Jacobi polynomials as spherical functions on rank one symmetric spaces of compact type, a number of novel nonlinear integral equations are presented. To validate the proposed method, some known nonlinear Volterra integral equations are solved and the series solutions converge to the known exact solutions. The convergence of the series solutions shows that the proposed power series method is accurate and reliable for solving such nonlinear Volterra integral problems.