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I do not know if this problem was considered before.

Prove that there are infinitely many pairs $(a,b)$ of mutually prime integers such that none of the polynomials

$P_n(x) = a*T_n(x) +b*U_n(x) $

has a quadratic factor of the form $(c*X^2+d)$ where $c$ and $d$ are nonzero integers. Here $T_n$ and $U_n$ are the Chebyshev polynomials of first and second kind and $n$ varies over the set of positive integers.

I suspect much more is true, that is, there are infinitely many such pairs $(a,b)$ for which $P_n(x)$ is "almost" irreducible for every $n$. The word "almost" is needed since both $T_n(x)$ and $U_n(x)$ have a factor of x when n is odd.

I did some testing in MAPLE and it looks like the set of such good pairs $(a,b)$ may even have positive density in $\mathbb{Z}^2$. Any help would be greatly appreciated.

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