Mathematics – General Mathematics
Scientific paper
2007-11-12
Mathematics
General Mathematics
15 pages
Scientific paper
Other than any odd prime whose factor is contained by the given even number 2N, the odd primes within open interval (1, 2N - 1) were defined as effective primes of 2N. Let 2N - pi ={\alpha}i Eq.s({\alpha}i); Product Pm ={\alpha}1{\alpha}2{\alpha}3 ...{\alpha}m-1{\alpha}m; Product Qm = p1p2p3 ...pm-1pm; Cm is the common divisor between Pm and Qm. where, 2N is any even number greater than 6; i = 1, 2, 3, ..., m and m \geq 2; pi is an effective prime of 2N, respectively, and pm is the maximum one among them. It was discussed that the following two equations always simultaneously hold when m \geq 2: [Pm - (-1)mQm]/2N = KmCm Pm/Cm - (-1)mQm/Cm = 2KmN where, Km is an integer. Based on factorial analysis on distributions of effective prime factors pi (i = 1, 2, 3, ..., m and m \geq 2) over these two equations, by using two-part method starting from m = 2, the following key theorem was obtained: Among Eq.s({\alpha}i), there is always at least one pair consisting of two identical equations when m \geq 2 as long as product Pm doesn't contain any factor other than factors pi(i = 1, 2, 3, ..., m and m \geq 2). According to this key theorem, it was verified that any given even number 2N greater than 6 can be represented as a sum of its two effective primes.
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