= Local solubility of the cubic x cubed minus x minus one
{title2=$y^2=x^3-x-1\text{ over }\mathbb Z_p\iff p\neq3$}
Modulo three the right side is always minus one, a nonsquare. At two, $x=-2$ gives the square <2-adic integer> $-7\equiv1\pmod8$. At twenty-three the same value is a nonzero square modulo the <prime> and lifts by <Hensel's lemma>. At every other <prime> at least five, the model has <good reduction of an elliptic curve> and the <Hasse bound> gives more than its single point at infinity, hence an affine smooth point. Lifting that point gives <p-adic integer> coordinates.
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