Abstract
In this work, we obtain bounds for the sum of the integer solutions of quadratic polynomials of two variables of the form <img src=image/13423078_01.gif> where <img src=image/13423078_02.gif> is a given natural number that ends in one. This allows us to decide the primality of a natural number <img src=image/13423078_02.gif> that ends in one. Also we get some results on twin prime numbers. In addition, we use special linear functionals defined on a real Hilbert space of dimension <img src=image/13423078_03.gif> , in which the relation is obtained: <img src=image/13423078_04.gif>, where <img src=image/13423078_05.gif> is a real number for <img src=image/13423078_06.gif>. When <img src=image/13423078_07.gif> or <img src=image/13423078_08.gif>, we manage to address Fermat's Last Theorem and the equation <img src=image/13423078_09.gif>, proving that both equations do not have positive integer solutions. For <img src=image/13423078_10.gif>, the Cauchy-Schwartz Theorem and Young's inequality were proved in an original way.
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