## Additive structure of *Z* mod (product of the first *k* primes),

with carry extension to integer prime pair sums (Goldbach)

### Nico F. Benschop

**Abstract** :
The product m_k of the first k primes (2 .. p_k) has
neighbours m_k \pm 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 multiplication mod m_k .
Squarefree modulus yields Z_{m_k} as disjoint union of
2^k groups, with as many idempotents -one per divisor of
m_k , forming a Boolean lattice BL. 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 . Hence 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).

The prime units in G_1(k) have principle (natural) values in the
corresponding set G(k) of naturals u < m_k. A proof by contradiction
and finite reduction using epimorphism G_1(k+1) \rightarrow G_1(k) mod m_k,
and verifying GC for 4< 2n < 30 (k=3), yields a contradiction for k=3.
Combined with Bertrand's postulate this proves GC: Each 2n \gt 4 is the sum
of two odd primes.

The structure of G_1(k) mod m_k is illustrated by the next features.
The smallest composite unit in G_1(k) mod m_k is (p_{k+1})^2 so its units
between p_{k+1} and (p_{k+1})^2 are all prime if considered as naturals
(their principle values in set G(k)), to be used as summands for successive
2n < m_k. For k=3 (m_3=30) it is shown by complete inspection that each
2n with 4< 2n < 30 is indeed the sum of two odd primes. For k>3 the addition
to obtain 2n < (p_{k+1})^2 produces no carry, thus yielding a natural sum.
The known Bertrand Postulate: p_{k+1} < 2p_k, implies overlapping intervals
for successive 2n.

**Keywords** : Residue arithmetic, multiplicative semigroup,
squarefree modulus, lattice of groups, additive structure, primesum,
primesieve, Goldbach conjecture.

Additive structure of Z(.) mod m_k (squarefree), and Goldbach's Conjecture.
(march 2012) . . . (10 pgs)

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intro (.htm)The structure of semigroups Z(.) mod m_k, and arithmetic with carry.