Abstract
In his work on Diophantine equations of the form y 2= ax 4+ bx 3+ cx 2+ dx+ e, Fermat introduced the notion of primitive solutions. In this expository note we intend to interpret this notion more geometrically, and explain what it means in terms of the arithmetic of elliptic curves. The specific equation y 2= x 4+4 x 3+10 x 2+20 x+1 was used extensively by Fermat as an example. We illustrate the nowadays available theory and software for studying elliptic curves by completely describing the rational solutions to this equation. It turns out that the corresponding Mordell-Weil group is free of rank 2; we obtain generators of this group.
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