THE DIOPHANTINE EQUATION FROM THE EYE OF PHYSICIST

Abstract

A Diophantine equation is an equation with integer coefficients, the solutions of which must be found among integers. The equation is named after the mathematician Diophantus of Alexandria (III century). Despite its simplicity, a Diophantine equation may have a nontrivial solution (several solutions) or has no solution at all. Fermat's Last Theorem and Pythagorean Theorem are the Diophantine equations too. In 1900 The German mathematician David Hilbert formulated the Tenth problem. After 70 years, the answer turned out to be negative: there is no general algorithm. Nevertheless, for some cases, schoolchildren can understand whether a Diophantine equation is solvable without resorting to calculations, relying on the methods of physics, symmetry and set theory.