33 papers
This paper introduces a complexity-based measure called "intelligence" to evaluate the quality of approximations of real numbers within a specific model, demonstrating its consistency with classical rational Diophantine approximation while posing an open question regarding the universal existence of such intelligent approximations.
This paper claims to prove the twin prime conjecture by establishing a lower bound for the count of twin primes up to using a new general method for estimating correlations of arithmetic functions.
This paper establishes a sufficient condition for an isomorphism between finite-order real matrix spaces and bivariate polynomial subspaces by linking a proposed polynomial problem to Lagrange bivariate interpolation, thereby proving the existence and uniqueness of solutions and providing construction formulas for these polynomials on finite rectangular grids.
This paper introduces the theory of the Collatz process and the method of dynamical balls to formulate the Collatz conjecture, explore its connection to Sophie Germain primes, and develop tools for analyzing the convergence of sequences generated by iterating on fixed integers.
This paper compiles a collection of polynomial Diophantine equations that are deceptively simple to formulate yet remain notoriously difficult to solve, highlighting significant open problems in the field.
This paper establishes a connection between the fixed points of the Josephus function and the Chinese Remainder Theorem, then utilizes a non-standard fractional base $3/2$ expansion to identify digit patterns that enable a recursive procedure for determining these fixed points.
This paper analyzes the finite sampling map for free-tail canonical systems, establishing that while local identifiability and inversion are achievable for specific finite-dimensional families via a weighted Fourier-Laplace linearization, the full free-tail class suffers from nontrivial first-order invisible directions at the free Hamiltonian that preclude any local inverse-Lipschitz estimate.
This paper provides a self-contained analytic proof of the continued fraction identity by transforming the classical Gauss continued fraction for evaluated at and demonstrates its super-exponential convergence advantage over the Gregory–Leibniz series.
This paper presents a new proof of the generalized Chapple-Euler relation for triangles inscribed in a circle and circumscribed about a central conic, establishing that the sum of the squared side lengths of such Poncelet triangles is invariant if and only if the circle is centered at either the conic's center or one of its foci.
This paper proposes the Wright-Euler Mersenne Exponent Hypothesis, which utilizes Euler's quadratic polynomial combined with nearest-integer rounding to identify candidate exponents for Mersenne primes, demonstrating a significantly higher accuracy and lower error rate compared to exponential regression models while effectively narrowing the search space for GIMPS testing.
This paper investigates the analytic continuation and prime-restricted theory of arctanh sums , establishing their meromorphic extension with specific poles and zeros, deriving Laurent expansions and Mittag-Leffler decompositions, and proving the unconditional transcendence of their prime-restricted analogues via a -cancellation mechanism.
This paper employs the AlphaEvolve computational engine to explore three combinatorial problems—graph reconstruction, the Alon-Tarsi parity problem, and Rota's Basis Conjecture—by iteratively refining local correcting steps to generate explicit algorithms and structural patterns that serve as foundations for future traditional mathematical proofs.
This paper utilizes Markov models and Monte Carlo simulations to analyze how the average duration of Chutes and Ladders changes as die roll probabilities approach certainty and to evaluate the impact of six distinct strategies involving optional coin-flip moves on game length.
This paper introduces a unified recursive framework that constructs elliptic extensions of Clausen-type functions by replacing trigonometric seeds with Jacobi theta functions, thereby establishing a structural correspondence between circular and elliptic settings while organizing the hierarchy into a single analytic object.
This numerically driven study establishes a robust multi-scale framework demonstrating that the inverse eigenvalue loci of generalized Lucas sequences exhibit a striking low-distortion geometric and potential-theoretic correspondence with the boundary of the Mandelbrot set, revealing shared structural organization across geometric, harmonic, and statistical levels.
This paper introduces the concept of simple closed-curve magnetization and applies it to solve the Bellman lost in the forest problem under specific geometric conditions relating the hiker's position to the forest boundary.
This paper explores the structural parallels between power functions and monic Chebyshev polynomials to establish a refined local classification of integers modulo a prime based on two quadratic characters, thereby extending classical number-theoretic concepts like primality testing, pseudoprimes, and cryptographic protocols to the Chebyshev setting.
This paper introduces and compares two construction methods for fuzzy betweenness relations in KM-fuzzy metric spaces, demonstrating their equivalence and establishing that they satisfy specific four-point and five-point transitivity properties.
This paper proposes an adaptive filtering framework utilizing canonical systems with time-varying Hamiltonian matrices, which are updated via a gradient-based scheme to minimize error while ensuring stability through Lyapunov analysis and preserving structural properties via specialized numerical integration and projection techniques.
This paper constructs noncontinuous, strictly increasing, bisymmetric binary operations on real intervals using Cantor-type sets to demonstrate that continuity is not guaranteed for non-reflexive operations of a specific form, while also proving that such operations become continuous and quasi-arithmetic if they are reflexive at two points.