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This is an open problem.

Let $x,y,z$ be coprime integers (not necessarily pairwise coprime) and no proper subset sum of $\{x,y,z,-(x+y+z)\}$ is zero.

For a quadruple $(x,y,z,-(x+y+z))$ define the quality $$q(x,y,z,-(x+y+z))=\frac{\log{(\max(|x|,|y|,|z|,|x+y+z|))}}{\log(rad(xyz(x+y+z))}$$ where $rad$ is the radical.

We have quality $q= 3+o(1)$ given by $$x=1, y=3 (t - 1) t, z=(t - 1)^3, -(x+y+z)=-t^3$$ and at integers set $t=2^{large}$.

Can we get quality $3+C$ infinitely often for positive $C$?

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  • $\begingroup$ Not quite as good, but you can get $3-o(1)$ by arranging that $t-1$ and $t$ are a square and twice a square. $\endgroup$
    – Aaron Meyerowitz
    Commented Dec 13, 2020 at 0:11
  • $\begingroup$ @AaronMeyerowitz You can reduce the radical of general, possibly irreducible f(x) to x^(degree(f(x))-1) by solving f(x)=0 mod p^n. $\endgroup$
    – joro
    Commented Dec 13, 2020 at 9:14

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