Authors: Nicolae BRATU, Adina CRETAN
This paper has been updated and completed thanks to suggestions and critics coming from Dr. Mike Hirschhorn, from the University of New South Walles. We want to express our highest gratitude. The paper appeared in an abbreviated form [6]. The present work is a complete form. For the homogeneous diophantine equations:x2 + by2 + cz2 = w2 there are solutions in the literature only for particular values of the parameters b and c. These solutions were found by Euler, Carmichael, Mordell. They proposed a particular solution for this equation in [3]. This paper presents the general solution of this equation as functions of the rational parameters b, c and their divisors. As a consequence, we obtain the theorem that every positive integer can be represented as the sum of three squares, with at most one of them duplicated, which improves on the Fermat –Lagrange theorem
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[v1] 2014-12-26 16:11:05
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