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Diophantine Equations These are polynomial equations for which postive integer solutions are sought. Diophantus’s Arithmetica (250) consisted of solutions to one or two hundred Diophantine equations, but neither complete solution sets nor systematic solution techniques were pre- sented. In 1900, David Hilbert gave a list of 23 unsolved math problems he thought were especially important. These are referred to as “Hilbert’s Problems”. Hilbert’s 10th Problem was to come up with a general algorithm for determining whether a Diophantine equation has a solution. In 1970 Yuri Matiyasevich proved that there is no such algorithm. Fermat’s Last Theorem The Theorem: If n > 2 there are no numbers x , y , and z such that x n + y n = z n . Fermat wrote in the margin of his copy of Diophantus’ Arithmetica : “It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvelous
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