CONSTANT D: Transcendence, Irreducible Complexity, and Significance in Prime Number Theory

A article titled explores the enigmatic constant D, defined as the binary number 0.0110101000101…where the k-th digit is 1 if and only if k is a prime number. This constant bridges discrete number theory and mathematical analysis, offering fresh insights into the chaotic yet deterministic nature of primes. Key Contributions: - Transcendence Hypothesis: The article argues for the transcendence of D, leveraging the nonlinear structure of primes, infinite systems of equations, and analogies to constants like Copeland-Erdős. Key lemmas demonstrate the non-overlapping layers of prime multiples and the aperiodicity of D, challenging algebraic descriptions. - The D Family: Introduces a novel concept where modifications to D (e.g., removing digits or shifting primes by constants) generate new numbers with similar transcendental properties. This family could reveal hidden correlations in prime distributions. - Applications: Highlights potential uses in harmonic analysis (e.g., Fourier transforms of D) and cryptography, where D-based algorithms might leverage prime unpredictability for secure key generation. - Connections to Major Hypotheses: Suggests links between D and the Riemann Hypothesis, proposing that D’s properties could illuminate the zeros of the zeta function. Methodology: The study employs binary sieves to construct D, supported by theorems such as the Prime Number Theorem and Dirichlet’s Theorem. The author’s prior work on "Binary Sieves" (cited in the references) underpins the technical framework. Implications: This work deepens our understanding of primes’ intrinsic complexity and opens pathways for interdisciplinary research. Whether D is proven transcendental or not, its study challenges mathematical paradigms and enriches both theoretical and applied domains. For researchers in number theory, cryptography, or mathematical analysis, this article is a compelling read that redefines the interplay between discreteness and continuity in mathematics.