= Local isotropy of the five seven thirteen form
{title2=$5x^2+7y^2+13z^2$}
This <ternary quadratic form> has a nonzero zero over $\mathbb Q_p$ exactly when $p\ne7$. Away from $2,5,7,13$, the finite-field square-set intersection proof supplies a nonsingular residue zero, and <Hensel lemma> lifts it. At 5 use $(0,1,1)$ modulo 5; at 13 use $(3,1,0)$. At 2 hold $x=2,z=1$ and apply the <strong form of Hensel lemma> to $7Y^2+33$ at $Y=1$: the function and derivative have <valuations> 3 and 1. At 7, the equation modulo 7 forces $x=z=0$ because 5 is not a square; reduction after dividing by 7 then forces $y=0$. Any normalized integral nonzero solution would therefore have all coordinates divisible by 7, a contradiction.
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