Additive structure of Z(.) mod m_k (squarefree) and Goldbach's Conjecture

Details Additive structure of Z(.) mod m_k (squarefree) and Goldbach's Conjecture Additive structure of Z(.) mod m_k (squarefree) and Goldbach's Conjecture

2001-03-14

arxiv.org/abs/math/0103091v5

Mathematics

General Mathematics

V1: 16 pages; Non-minimal expository style. Application of semigroups to arithmetic. See also http://www.iae.nl/users/benschop

Scientific paper

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 i1) 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.

Affiliated with

Also associated with

No associations

Rating

Additive structure of Z(.) mod m_k (squarefree) and Goldbach's Conjecture does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Additive structure of Z(.) mod m_k (squarefree) and Goldbach's Conjecture, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Additive structure of Z(.) mod m_k (squarefree) and Goldbach's Conjecture will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-175642