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Smallest solutions for various sequences of integer (or so-called Diophantine) equations. indicates that it can be proved that no solution exists. A blank indicates that I know only that no solution exists below a billion. Methods for resolving some of the equations in the first column were known in antiquity; all had been resolved by the 1800s. Practical methods for resolving the so-called elliptic curve equations in the second column were developed only in the 1980s. No general methods are yet known for most of the other equations given—and some classes of them may in fact show undecidability.

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From Stephen Wolfram: A New Kind of Science [citation]