| _version_ | 1866901721794478080 |
|---|---|
| author | Revista, Zen MATH, 10 |
| author_facet | Revista, Zen MATH, 10 |
| contents | This paper investigates the fine structure of Goldbach-Waring bases by examining their emergent Schnirelmann densities. The Goldbach-Waring problem, a powerful synthesis of Goldbach's conjecture and Waring's problem, deals with representing integers as sums of primes and powers of integers. While the existence of such bases is well-established through advanced analytic number theory, the specific quantitative measure of their density, particularly for initial segments of natural numbers, remains less explored. We define "emergent Schnirelmann densities" as the behavior of the Schnirelmann density function for increasing but finite upper bounds, contrasting it with purely asymptotic results. Our methodology combines rigorous analytical number theoretic techniques, including the circle method and sieve theory, with extensive computational verification for smaller values of integers. This dual approach reveals nuanced patterns in how these bases cover the natural numbers. We present theoretical bounds and empirical observations on how the number of prime summands and the degree of the powers influence the density, shedding light on the "fineness" of their coverage. The results provide new insights into the additive properties of integers and offer a quantitative lens through which to understand the distribution and combinatorial aspects of sums involving both prime numbers and perfect powers. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17835347 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Emergent Schnirelmann Densities: Unveiling the Fine Structure of Goldbach-Waring Bases Revista, Zen MATH, 10 This paper investigates the fine structure of Goldbach-Waring bases by examining their emergent Schnirelmann densities. The Goldbach-Waring problem, a powerful synthesis of Goldbach's conjecture and Waring's problem, deals with representing integers as sums of primes and powers of integers. While the existence of such bases is well-established through advanced analytic number theory, the specific quantitative measure of their density, particularly for initial segments of natural numbers, remains less explored. We define "emergent Schnirelmann densities" as the behavior of the Schnirelmann density function for increasing but finite upper bounds, contrasting it with purely asymptotic results. Our methodology combines rigorous analytical number theoretic techniques, including the circle method and sieve theory, with extensive computational verification for smaller values of integers. This dual approach reveals nuanced patterns in how these bases cover the natural numbers. We present theoretical bounds and empirical observations on how the number of prime summands and the degree of the powers influence the density, shedding light on the "fineness" of their coverage. The results provide new insights into the additive properties of integers and offer a quantitative lens through which to understand the distribution and combinatorial aspects of sums involving both prime numbers and perfect powers. |
| title | Emergent Schnirelmann Densities: Unveiling the Fine Structure of Goldbach-Waring Bases |
| url | https://doi.org/10.5281/zenodo.17835347 |