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comment Parametric solutions of Pell's equation
Also, you say that "for generic $a$, the Galois group of $T_n(x)=a$ should be dihedral of order $2n$" -- what I observe from GAP computations is that for $n \geq 5$ this doesn't seem to be true -- e.g. if $T_5(x)-a$ is irreducible, I find that its Galois group is the semidirect product of ${\rm C}_5$ and ${\rm C}_4$, and for $n = 7$ I find it's a semidirect product of ${\rm C}_7$ and ${\rm C}_6$, and for $n=8$ it is 'often' a group of order 32.
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comment Parametric solutions of Pell's equation
Furthermore, $T_4(x)-a$ is reducible in $\mathbb{Q}[x]$ for 151 among the first 1000 values of $a$, and $T_5(x)-a$ is reducible for 10 among the first 1000 values of $a$, and $T_6(x)-a$ is reducible for 311 among the first 1000 values of $a$, and $T_7(x)-a$ is reducible for 6 among the first 1000 values of $a$, and $T_8(x)-a$ is reducible for 151 among the first 1000 values of $a$.
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comment Parametric solutions of Pell's equation
I have checked the first 10000 nonsquares in that sequence, and found that for 1793 of them, $T_3(x)-a$ has a rational root. Thus the density seems roughly the same as for the first 60 which you checked. -- Do you still think the density drops to 0?
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