## Prime Sieve Using Multiplication Operation Table

**Authors:** Jongsoo Park

Ben Green and Terence Tao showed that for any positive integer k, there exist infinitely many
arithmetic progressions of length k consisting only of prime numbers. [14] Four parallel proofs of Szemer'edi's
theorem have been achieved; one by direct combinatorics, one by ergodic theory, one by hypergraph theory, and
one by Fourier analysis and additive combinatorics. Even with so many proofs, Professor T. Tao points out that
with this problem, there remains a sense that our understanding of this result is incomplete; for instance, none of
the approaches were powerful enough to detect progressions in the primes, mainly due to the sparsity of the
prime sequence. [22] Oliver Lonsdale Atkin introduced a prime sieve using irreducible binary quadratic forms
and modular arithmetic; the algorithm enumerates representations of integers by certain binary quadratic forms.
A way that uses modular arithmetic is widely known: 6n+δ, 12n+δ, 30n+δ, 60n+δ.[31] In this paper,
we assert that the composite number of the 12n+1, 5, 7, 11 series as selected by a Modular Arithmetic and
Multiplication Table are not random but consist of very structural and regular arithmetic progression groups.

**Comments:** 76 pages

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### Submission history

[v1] 15 Mar 2010

[v2] 16 Mar 2010

[v3] 2 Apr 2010

[v4] 16 Jul 2010

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