Additive structure of Z(.) mod m_k (squarefree) and Goldbach's Conjecture
Abstract
The product m_k of the first k primes (2..p_k) has neighbours m_k +/- 1 with all prime divisors beyond p_k, implying there are infinitely many primes [Euclid]. All primes between p_k and m_k are in the group G_1 of units in semigroup Z_{m_k}(.) of mutiplication mod m_k. Due to the squarefree modulus Z_{m_k} is a disjoint union of 2^k groups, with as many idempotents - one per divisor of m_k, which form a Boolean lattice BL. The generators of Z_{m_k} and the additive properties of its lattice are studied. It is shown that each complementary pair in BL adds to 1 mod m_k and each even idempotent e in BL has successor e+1 in G_1. It follows that G_1+G_1 \equiv E, the set of even residues in Z_{m_k}, so each even residue is the sum of two roots of unity, proving "Goldbach for Residues" mod m_k ("GR"). . . . Induction on k by extending residues mod m_k with "carry" a < p_{k+1} of weight m_k, yields a prime sieve for integers. Failure of Goldbach's Conjecture ("GC") for some 2n contradicts GR(k) for some k. By Bertrand's Postulate (on prime i<p<2i for each i>1) successive 2n are in overlapping intervals, while the smallest composite unit in G_1 mod m_k is p_{k+1}^2, yielding "GC": Each 2n > 4 is the sum of two odd primes.
Keywords
Cite
@article{arxiv.math/0103091,
title = {Additive structure of Z(.) mod m_k (squarefree) and Goldbach's Conjecture},
author = {N. F. Benschop},
journal= {arXiv preprint arXiv:math/0103091},
year = {2009}
}
Comments
V1: 16 pages; Non-minimal expository style. Application of semigroups to arithmetic. See also http://www.iae.nl/users/benschop/ng-abstr.htm and intro at http://www.iae.nl/users/benschop/fewago.htm Replaced: V3 is shorter version (10 pgs) and has last Thm(GC) with new last paragraph. V4: new sections 4 and 5. V5: extended 'Notation' paragraph in the Intro