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== Diophantine Equations The Pythagorean theorem states that if $a$ and $b$ are the altitude and base of a right triangle, then $a^2 + b^2 = c^2$, where $c$ is the hypotenuse. It was known in ancient times that $a,b,c = 3,4,5$ are the sides of a right triangle. This example leads to the question of whether the "Pythagorean equation" has other solutions. The answer is, _yes indeed_, as the result below asserts. [env.theorem%pythag-eq] -- The equation \[ a^2 + b^2 = c^2 \] has infinitely many non-proportional integer solutions. -- It is natural to ask about solutions to analogue of the Pythagorean equation, but with exponent higher than two. Pierre Fermat considered this problem and arrived at the following conclusion. [env.theorem#mu%fermat-eq] -- The only solution in the integers of the equation \[ a^3 + b^3 = c^3 \] are the obvious ones in which one of the variables is zero. -- These results constitute the foundation of the subject of Diophantine equations.
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