Sum-factor decompositions in rings and arithmetic applications I

Details Sum-factor decompositions in rings and arithmetic applications I Sum-factor decompositions in rings and arithmetic applications I

2009-12-22

arxiv.org/abs/0912.4412v2

Mathematics

Number Theory

33 pages; The abstract is slightly changed; the Remark 3.18 is added.

Scientific paper

In this paper, by introducing and constructing several new structures about the decomposition phenomenon in algebra, we study the sum-factor collapse property of an arbitrary ring. As an application, we study and analyze several classical problems in additive number theory by this new method. For example, we obtain that (see Theorem 3.17.(2)), for all sufficiently large integers $ n, $ the sum of any $ n $ number of odd integers $ (\geq 3) $ is equal to the sum of $ n $ number of odd primes. This is consistent with the prediction of Goldbach conjecture. Some further questions are also presented and discussed.

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