Abstract: This paper provides a analytical proof of the Riemann Hypothesis using a differential interaction operator Φ(s,δ) on the Hilbert space l²(N). By mapping the Dirichlet η-function to a trace-class operator representing the interaction between states shifted by ±δ from the critical line, we derive a Phase-Torque J(δ,t) governed by a hyperbolic sine bias. We establish a Product Criterion showing that the operator trace vanishes if and only if a zero exists at either 1/2 + δ + it or 1/2 − δ + it. Finally, we establish the convergence criteria for this operator and demonstrate that the Diophantine independence of prime logarithms, amplified by the hyperbolic lever, prevents the trace from vanishing off the critical line.

Abstract: In this article, the usual factorials and binomial coefficients have been generalized and extended to negative integers. Based on this generalization and extension, a new kind of polynomials has been proposed, which has directly led to the non-classical hypergeometric orthogonal polynomials and the non-classical second-order hypergeometric linear ordinary differential equations. The resulting polynomials can be used in non-relativistic and relativistic quantum mechanics, particularly in the case of the Schrödinger equation and Dirac equations for an electron in a Coulomb potential field.

Abstract: We prove the irrationality of a family of normalized odd zeta values of the form \( \dfrac{\zeta(2n+1)}{\pi^{2n+1}},\,n\in\mathbb{N},\,n\geq 3. \) Our approach is based on constructing explicit integer linear forms in the quantities \( I_n=4(4^n-1)\left[\dfrac{\zeta(2n)\zeta(2n+2)}{\zeta(2n+1)^2}-1\right]-1 \), and applying a refinement of Dirichlet's approximation theorem. The construction of the \( I_n \) is probabilistic in origin. We prove that the sequence of denominators produced by successive rational approximations yields infinitely many nontrivial integer relations of the type \( \Lambda_m^{(q)}=A_m^{(q)} I_n-B_m^{(q)}, \) with \( |\Lambda_m^{(q)}| \) (\( q \) being a parameter) decaying towards zero as \( m \) approaches infinity. This permits us to invoke a general irrationality criterion and thereby deduce that \( I_n \) is irrational for each \( n\geq 3 \). Consequently, each corresponding normalized odd zeta value is irrational. Our method combines ideas from probability theory, analytic combinatorics and Diophantine approximation, and complements earlier work of Apéry, Beukers, Rivoal, and Zudilin.

Abstract: Let \( \mathcal{X} \) be a p-adic Hilbert space. Let \( A: \mathcal{D}(A)\subseteq \mathcal{X}\to \mathcal{X} \) and \( B: \mathcal{D}(B)\subseteq \mathcal{X}\to \mathcal{X} \) be possibly unbounded self-adjoint linear operators. For \( x \in \mathcal{D}(A) \) with \( \langle x, x \rangle =1 \), define \( \Delta _x(A):= \|Ax- \langle Ax, x \rangle x \|. \) Then for all \( x \in \mathcal{D}(AB)\cap \mathcal{D}(BA) \) with \( \langle x, x \rangle =1 \), we show that (1)max{Δx(A),Δx(B)}≥|⟨[A,B]x,x⟩2+(⟨{A,B}x,x⟩−2⟨Ax,x⟩⟨Bx,x⟩)2||2| and (2)max{Δx(A),Δx(B)}≥|⟨(A+B)x,y⟩|,∀y∈X satisfying ‖y‖≤1,⟨x,y⟩=0. We call Inequality (1) as p-adic Heisenberg-Robertson-Schrodinger uncertainty principle and Inequality (2) as p-adic Maccone-Pati uncertainty principle.

Abstract: This paper undertakes a foundational exploration of the nature of mathematics from both historical and philosophical perspectives, with a primary focus on the Indian intellectual tradition. It traces the evolution of mathematical thought from ancient Vedic texts such as the ´Sulba S¯utras, through the formal grammar of P¯an. ini, to modern abstract mathematics including group theory, automata, and topology. The investigation is rooted in the dual inquiries of ontology and epistemology, examining what it means for mathematics to be and how mathematical knowledge is constructed and validated. Particular emphasis is placed on the Indian concepts of gan. ita (mathematics), ´s¯unya (zero), and ´s¯unyat¯a (emptiness), and their correspondence with Western notions such as the Cartesian dualism, the set-theoretic empty set, and symbolic logic. The paper explores the recursive cosmological cycles found in Indian time theory, mathematical cosmology, and ritual geometry, showing how these ideas anticipated or paralleled developments in modern mathematics, including measure theory, combinatorics, and fractals. With detailed references to logical systems (Ny¯aya), sacred architecture (v¯astu-´s¯astra), cyclic time constructs (kalpas and yugas), and formal structures in linguistic grammar (As.t. ¯adhy¯ay¯ı), the paper argues for a view of mathematics as both a sacred science and a system of abstract formalism. Across these investigations, mathematical structures are treated not merely as tools for calculation but as profound reflections of metaphysical principles, visualizable through mandalas, yantras, and cosmological diagrams. This study invites a reassessment of how different cultures have understood and visualized mathematics as an expression of cosmic and cognitive order.

Abstract: Around 1637, Pierre de Fermat famously wrote in the margin of a book that he had a proof for the equation aⁿ + bⁿ = cⁿ having no positive integer solutions for exponents n > 2. While Andrew Wiles provided a complete proof in 1994 using advanced 20th-century machinery, the question of whether a simpler proof exists remains a subject of intense mathematical interest. In this work, we focus on a significant restricted case of the theorem: the situation in which the exponent n possesses a prime divisor p that does not divide the quantity abc. Under this natural arithmetic condition, we develop an elementary argument—based on Barlow’s Relations and p-adic valuations—that leads to a contradiction. These methods lie closer to the classical number-theoretic framework that Fermat himself might have envisioned, and they illuminate structural features of the Fermat equation that persist across related Diophantine problems.

Abstract: The advances related to Modular Elliptic Curves and the Fermat – Wiles’ theorem in recent years has opened the door to entirely new approaches to numerous problems and techniques. This paper studies the linking between a family of exponential function and the Fermat – Wiles’ theorem. In this family, the convergence of the coordinates of the abscissas of the maxima and the equation determined by the fulfilment of the maximum condition leads to a singular connection with the Fermat – Wiles’ theorem.

Abstract: Massera and Schaffer [\textit{Ann. Math. (2), 1958}] derived a breakthrough upper bound for the Clarkson angle between two nonzero vectors in a normed linear space, which was later improved by Maligranda [\textit{Am. Math. Mon., 2006}]. Pecaric and Rajic [\textit{Math. Inequal. Appl., 2007}] extended Maligranda's inequality to finitely many nonzero vectors. We derive a non-Archimedean version of Massera-Schaffer-Maligranda-Pecaric-Rajic inequality.

Abstract: In 1992, Hudzik and Landes derived a breakthrough generalization of the triangle inequality for two nonzero elements in normed linear spaces, which was generalized to finitely many nonzero elements independently in 2006 by Dragomir and in 2007 by Kato, Saito and Tamura. We derive a non-Archimedean version of Hudzik-Landes-Dragomir-Kato-Saito-Tamura inequality.

Abstract: We demonstrate a new quantitative method to the sieve of Eratosthenes, which is an alternative to the sieve of Legendre. In this method, every element of a given set is sifted out once only; and therefore, this method is free of the Mobius function and of the parity barrier. Using this method, we prove that every sufficiently large even number is the sum of two primes, and that every even number is the difference of two primes in infinitely many ways.

Abstract: Let \(x_1,\dots,x_n\in[-1,1]\) be distinct nodes and let \[ l_k(x)=\prod_{i\neq k}\frac{x-x_i}{x_k-x_i} \] denote the associated Lagrange interpolation polynomials. Erd\H{o}s posed the problem of minimizing the functional \[ I(x_1,\dots,x_n)=\int_{-1}^1 \sum_{k=1}^n |l_k(x)|^2\,dx \] and determining its asymptotic behavior as \(n\to\infty\). It was known that \[ 2-O\!\left(\frac{(\log n)^2}{n}\right)\le \inf I \le 2-\frac{2}{2n-1}, \] with the upper bound attained by nodes related to Legendre polynomials.In this paper, we develop a variational framework based on Christoffel functions, orthogonal polynomial asymptotics, and entropy methods to resolve this problem asymptotically. Our main contributions are:\begin{enumerate} \item[(i)] We prove that any asymptotically minimizing sequence of nodes must equidistribute with respect to the arcsine measure on \([-1,1]\). \item[(ii)] We establish a sharp \(O(1/n)\) lower bound, improving the longstanding \(O((\log n)^2/n)\) result of Erd\H{o}s--Szabados--Varma--V\'ertesi. \item[(iii)] We identify that the leading correction arises from microscopic endpoint regions and formulate an \emph{entropy rigidity hypothesis} connecting deterministic minimization to equilibrium log-gas behavior. \item[(iv)] Under a conjectured \emph{endpoint universality} principle for discrete Christoffel functions, we derive the first-order asymptotic expansion \[ \inf I = 2 - \frac{c}{n} + o\!\left(\frac{1}{n}\right), \] with an explicit constant \(c>0\) expressed via the Airy kernel. \item[(v)] We show that the Legendre--integral nodes are asymptotically optimal and rigid, and support all theoretical predictions with detailed numerical experiments, including verification of edge rigidity and Airy-type endpoint scaling. \end{enumerate}The expansion in (iv) is conditional on an endpoint universality conjecture (Conjecture~5.1), whose rigorous proof remains an open problem. A complete verification would finalize the asymptotic solution of Erd\H{o}s's interpolation extremal problem and establish a deeper connection to universality in random matrix theory.

Abstract: This work offers formal proofs which were enabled by a change in perspective from studying individual integer iterations to analyzing how the conjecture's rules organize the positive integers. The proofs rigorously demonstrate the satisfaction of several critical criteria: the universal inclusion of all positive integers within the proof's scope; the disclosure of a simple and predictable pattern among the numbers; the conclusive absence of any major loops; the demonstration that no number continuously increases indefinitely without eventually decreasing; and the ultimate convergence of all positive integers to 1 when subjected to the Collatz iteration rules. Formal verification of these proofs was conducted using the Isabelle/HOL proof assistant.

Abstract: In this article, we prove that for every integer \(n \geq 2\), there exist positive integers \(t\) and \(x\) such that the expression \( E = t^2(4x - n)^2 - 2xtn \) is always a perfect square.

Abstract: In this paper we first give a new formula of the Liouville function and then by using the method for proving the Bateman-Horn conjecture, we give a parallel proof of the Chowla conjecture.

Abstract: We demonstrate that the fine structure constant alpha⁻¹ ≈ 137.036 emerges necessarily from the deepest mathematical structure of reality: the zeros of the Riemann zeta function zeta(s). We present an exact formula connecting alpha⁻¹ to the first four nontrivial zeros gamma_1, gamma_2, gamma_3, gamma_4 of zeta(1/2 + it). The derivation combines spectral theory of magnetic Schrodinger operators on hyperbolic surfaces, the Selberg-Gutzwiller trace formula, and arithmetic geometry. The resulting value matches the experimental CODATA 2018 value with precision 2.7 × 10⁻¹³. This establishes a profound connection between number theory and fundamental physics.

Abstract: Nica and Sprague [\textit{Am. Math. Mon., 2023}] derived a non-Archimedean version of the Gershgorin disk theorem. We derive a non-Archimedean version of the oval (of Cassini) theorem by Brauer [\textit{Duke Math. J., 1947}] which generalizes the Nica-Sprague disk theorem. We provide applications for bounding the zeros of polynomials over non-Archimedean fields. We also show that our result is equivalent to the non-Archimedean version of the Ostrowski nonsingularity theorem derived by Li and Li [\textit{J. Comput. Appl. Math., 2025}].

Abstract: There are added some matters for the estimation of \( H(n,m) \) in the appendix.

Abstract: Let 1 < a1 < a2 < · · · be integers with \( \sum_{k=1}^\infty a_k^{-1}<\infty \), and set \( F(s)=1+\sum_{k=1}^\infty a_k^{-s}, \qquad \Re s>1. \) A question of Erdős and Ingham, recorded as Erdős Problem #967 in a compilation by T. F. Bloom (accessed 2025--12--01), asks whether one always has \( F(1+it)\neq 0 \) for all real t. This paper does not resolve the problem; instead, it develops a modern dynamical-systems framework for its study. Using the Bohr transform, we realise $F$ as a Hardy-function on a compact abelian Dirichlet group and interpret \( F(1+it) \)as an observable along a Kronecker flow. Within this setting we establish a quantitative reduction of the nonvanishing question to small-ball estimates for the Bohr lift, formulated as a precise conjecture, and we obtain partial results for finite Dirichlet polynomials under Diophantine conditions on the frequency set. The approach combines skew-product cocycles, ergodic and large-deviation ideas, and entropy-type control of recurrence to small neighbourhoods of -1, aiming at new nonvanishing criteria on the line \( \Re s=1 \).

Abstract: The Bateman-Horn conjecture is a conjecture on prime values in polynomials. We prove it by Golomb's method.

Abstract: In this article we show that the polynomial \( t^2(4x - n)^2 - 2ntx \) does not always admits a perfect square with \( n\geq 2 \) and \( (x,t)\in \mathbb{(N^*)^2} \). We prove this when \( n=3 \) and we show by contradiction that one of x or t (in the expression \( t^2(4x - 3)^2 - 6tx \)) isn't an integer.