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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