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Let $k$ be positive integer, not a square and let $u_k,v_k$ be non-trivial solutions to the Pell equation $u_k^2-k^3 v_k^2=\pm 1$.

Q1 How small $u_k$ can be infinitely often as function $k$?

This gives rise to the good abc triple $(u_k^2,-k^3 v_k^2,\mp 1)$ so abc implies $u_k$ can't be too small, it can't be $\log(u_k)=C\log(k)$.

$k/\log(u_k)$ appears to diverge.

$\log^2(k^3) / \log(u_k)$ appears better, with large sporadic peaks. $\log(k^3)/\log(u_k)$ doesn't appear to go to zero fast and also have sporadic peaks.

Q2 for positive constants $C_0, 1 \le C < 2$ do we have $u_k<C_0 \exp(\log^C(k^3))$

Possible approach is to find $\sqrt{k^3}$ with relatively short continued fraction expansion.

We believe the set of small $u_k$ to be very sparse, so the numerical evidence might be misleading.

$k$ is on the $x$ axis:

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  • $\begingroup$ @FedericoPoloni As $k$ varies I am computing $u$ and then plot log(k)/log(u), there is legend at top. $\endgroup$
    – joro
    Commented Sep 24, 2019 at 11:39
  • $\begingroup$ @FedericoPoloni Yes, $k$ is on $x$ axis, sage doesn't label them. I think I am missing some $k$ because $u$ is rather large, will try to make it better. $\endgroup$
    – joro
    Commented Sep 24, 2019 at 13:07
  • $\begingroup$ The classic paper on abc, by J.Browkin-J.Brzeziński, has applied continued fractions. $\endgroup$
    – Wlod AA
    Commented Sep 25, 2019 at 22:36

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