Irrational number?

Question

Is the solution of the equation

$$x + \arctan(x) = \pi$$

irrational ?

The equation of $x + \arctan(x) = 1$ must be transcendental because for any nonzero algebraic $x$, $arctan(x)$ is transcendental, but this argument does not work for the above equation. The continued fraction of the above solution has more than $97000$ terms (PARI), so the answer seems to be yes. But can it be proven ?

Answer 1

Yes.

Let $x$ be the solution to $x+\arctan(x) = \pi$, then $$\arctan(x)=\pi-x \\ \Rightarrow x=tan(\pi-x) \\ \Rightarrow x=-tan(x).$$

Thus if $x$ would be rational, also $tan(x)$ would be rational. This is impossible: You can use the statement you gave for showing that $x+arctan(x)=1$ is irrational. Here is another reference:

Prove that if $x$ is a non-zero rational number, then $\tan(x)$ is not a rational number and use this to prove that $\pi$ is not a rational number.