Number Theory

    

Opening the Sieve of Eratosthenes, and Some Theorems on the Distribution of Primes

Authors: Izzie Boxen

The Sieve of Eratosthenes is taken as a definition of primes and is examined in a way that "opens" it into an array of rows labeled as primes and columns labeled as numbers. Through the introduced concept of prime candidates, numbers in each row that have the potential of being declared primes in lower rows, the opened Sieve reveals repeating and inter-related patterns of these prime candidates as well as other defined entities. These allow development of a number of relations that prove useful in examining the distribution of primes, with some new theorems being proved. Included are four proofs of a corollary used to prove Bertrand’s postulate, two proofs for Lim((p_[n+1]-p_n)/p_n)=0, and proofs of conjectures by Brocard, Legendre, Andrica, and Oppermann.

Comments: 69 Pages.

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Submission history

[v1] 2025-07-19 01:35:53
[v2] 2025-07-31 20:32:09
[v3] 2025-10-06 03:23:01
[v4] 2025-12-31 02:45:26
[v5] 2026-06-06 00:20:35

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