V.29 Rational Points on Curves and the Mordell Conjecture

Suppose that we wish to study a Diophantine equation such as x3 + y3 = z3. A simple observation we can make is that studying integer solutions to this equation is more or less equivalent to studying rational solutions to the equation a3 + b3 = 1: indeed, if we had integers x, y, and z such that x3 + y3 = z3, then we could set a = x/z and b = y/z and obtain rational numbers with a3 + b3 = 1. Conversely, given rational numbers a and b with a3 + b3 = 1, we could multiply a and b by the lowest common multiple z of their denominators and set x = az and y = bz, obtaining integers x, y, and z such that x3 + y3 = z3.

The advantage of doing this is that it reduces the number of variables by 1 ...

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