The diophantine equation $T_{n}(x)=a$

Question

Let us consider the Chebychev polynomial function $T_{n}(x)$ where $x$ is a positive integer variable and consider the diophantine equation $$T_{n}(x)=a......(*)$$ where $a$ is a positive integer. I am asking if there exist some results on the current literature concerning the integer solutions of $(*)$. I am interested essentially in the case where $a$ is a prime number.

Answer 1

The question is already difficult for $T_n(x)$ for small $n\ge 2$. For example, we have the Diophantine equation $$ T_2(x)=2x^2-1=p. $$ It is conjectured that this has a solution for infinitely many primes $p$.

There are many general results and conjectures concerning integer solutions of $f(x)=p$ for a polynomial $f\in \Bbb Z[x]$ and a prime $p$.

1. The Bunyakovsky conjecture gives a criterion for a polynomial $f(x)$ in one variable with integer coefficients to give infinitely many prime values in the sequence $f(n)$ for $n=1,2,3\ldots $

2. Some irreducible polynomials do not represent any prime at all. For example $f(x)=x(x+1)+4$ is always even, but doesn't represent $p=2$. For more details and a lot of interesting references see this post:

Polynomials representing primes