Suppose that we wish to study a Diophantine equation such as *x*^{3} + *y*^{3} = *z*^{3}. 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 *a*^{3} + *b*^{3} = 1: indeed, if we had integers *x*, *y*, and *z* such that *x*^{3} + *y*^{3} = *z*^{3}, then we could set *a* = *x/z* and *b* = *y/z* and obtain rational numbers with *a*^{3} + *b*^{3} = 1. Conversely, given rational numbers *a* and *b* with *a*^{3} + *b*^{3} = 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 *x*^{3} + *y*^{3} = *z*^{3}.

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

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