A proof of the strong Goldbach conjecture

• Autores: Paul S. Bruckman
• Localización: International journal of mathematical education in science and technology, ISSN 0020-739X, Vol. 39, Nº. 8, 2008, págs. 1102-1109
• Idioma: inglés
• DOI: 10.1080/00207390802136560
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• Resumen

  • An elementary proof of the 'strong' version of Goldbach's Conjecture (GC) is presented. Letting d(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 ?(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.