Valuations, arithmetic progressions, and prime numbers
Shin-ichiro Seki
TL;DR
This paper presents new proofs for the infinitude of primes using valuation theory and offers a novel proof of the divergence of the sum of prime reciprocals by leveraging Roth's theorem and Euler-Legendre's theorem for arithmetic progressions.
Contribution
It introduces two valuation-based proofs of prime infinitude and a new proof of the divergence of prime reciprocals sum using advanced number theory theorems.
Findings
Two valuation-based proofs of the infinitude of primes
A new proof of the divergence of the sum of prime reciprocals
Application of Roth's and Euler-Legendre's theorems to prime number theory
Abstract
In this short note, we give two proofs of the infinitude of primes via valuation theory and give a new proof of the divergence of the sum of prime reciprocals by Roth's theorem and Euler-Legendre's theorem for arithmetic progressions.
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Taxonomy
TopicsAnalytic Number Theory Research · Algebraic Geometry and Number Theory · Limits and Structures in Graph Theory