International journal of mathematical education in science and technology
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Volume 39 (2008) nr. 8 pages 1102-1109
Year:
2008-12
Contents:
An elementary proof of the 'strong' version of Goldbach's Conjecture (GC) is presented. Letting δ(k) represent the characteristic function of the odd primes, our proof utilizes a theorem previously derived by the author, a modification of which allows us to estimate the function f(u) ≡ [image omitted], where 0 < u < 1, in terms of the integral g(u) ≡ [image omitted]. In turn, g(u) is estimated in terms of a power series h(u) ≡ [image omitted]. With this result, it is then shown that f(u) is greater than u3(1 - u2)-1/2, which implies that f(u) = u3(1 - u2)-c/2 for some c = c(u) ∈ (1, 2). Squaring f(u) and by comparing coefficients, we conclude that the Goldbach function θ(2N) ≡ [image omitted], the counting function of the number of all permutations of odd primes p and q such that p + q = 2N, is at least equal to one; this is the 'strong' form of the Goldbach conjecture.
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