Is it possible to know if two polynomials will eventually have an equal value?

Question

Is it possible to know that the polynomial $P_1(x_1) = P_2(x_2)$, if so how to find the value that they both equal?

Example: $$(13)^2 = 3(7)^2 + 3(7) + 1 = 169$$

Answer 1

If you restrict to integers, this problem is quite arduous.

To illustrate, it took 158 years to prove that the only powers that differ by one are $3^2$ and $2^3$.

Also, the smallest solution of the "innocent"

$$x^2=109y^2+1$$ is known to be $x=158070671986249,y=15140424455100 $ (but $x^2=110y^2$ is solved by $x=21,y=2$). This tends to show that a brute-force search is hopeless.

AFAIK, there is no general method to tackle your problem. (Without the integer constraint, it is much more accessible.)