$\pi + e$ is rational or $\pi-e$ is rational

I was asked to find the truth value of the statement:

$$ \pi + e \; \text{ is rational or } \pi - e\; \text{ is rational } $$

I am only allowed to use the fact that $\pi, e $ are irrational numbers and cannot use the theory of transcendental numbers.

Cannot proceed. any help would be appreciated.

Solution 1:

This seems to be an open problem. It is a conjecture that the statement is false, i.e. that $\pi + e$ and $\pi - e$ are irrational. According to Wikipedia this remains unproven. (Just imagine the impact of the discovery of an equation such as $\pi=e+\frac{4233108252.........3123782}{31238295213.......0591231}$ ... unbelievable!)

Remark that at least one of those numbers is irrational, even transcendental (but this doesn't prove that both are irrational!). For if both would be algebraic, then their sum would be algebraic, which is $2 \pi$, a contradiction. Notice that this argument is not constructive at all, and again that it does not decide if "$\pi+e$ is rational or $\pi-e$ is rational" is false or not, it only proves that the stronger statement "$\pi+e$ is rational and $\pi-e$ is rational" is false.

Solution 2:

$2\sqrt2$ and $\sqrt2$ are two distinct irrational numbers s.t. the statement: '$2\sqrt2+\sqrt2$ is rational or $2\sqrt2-\sqrt2$ is rational' is not true.

$\sqrt2$ and $2-\sqrt2$ are two distinct irrational numbers s.t. the statement: '$\sqrt2+(2-\sqrt2)$ is rational or $\sqrt2-(2-\sqrt2)$ is rational' is true.

This illustrates that the statement cannot be proved purely based on the fact that $\pi$ and $e$ are irrational.