What mathematical consequences might there be if Euler Mascheroni constant is rational?

Question

So far as I know, no one has proved the irrationality of Euler Mascheroni constant. There are discussions about the difficulty of proving the irrationality of this constant.

Since we cannot prove that this constant is irrational, is it not theoretically possible that this number is actually rational? Perhaps it can be written as the form $p/q$ where $q$ is a huge integer (far beyond the capacity of all current supercomputers). If so, it sounds to me like a prank on mathematicians from God.

Intuitively I also believe that this constant should be irrational; but isn't it also (perhaps extremely remotely) possible that it is rational? If so, all current effort in proving its irrationality is in the wrong direction.

Edit: Being reminded by a comment, I am basically asking "if it is rational, how does it affect mathematics"? For example, are there any theories based on the irrationality of this constant that need to be overthrown?

Answer 1

Since nobody ever proved whether $\gamma$ is rational or irrational, then of course that it is possible that it is rational. However, since that is very unlikely it is natural that people try to prove that it is irrational. In the end, if it turns out to be rational, then a lot of effort will have been wasted, in the sense that those who did that research were trying to prove something that cannot be proved, but that's how research is.

Answer 2

It's at least conceivable that, even if $\gamma$ is rational, an attempted proof of its irrationality could help to find that out. For example, instead of proving $\gamma\neq\frac{p}{q}$ unconditionally, you might have a proof that works unless $q=3^{5+2^n}$. Armed with that hint, you might actually be able to prove that such an $n$ exists.

Answer 3

To answer the question about the consequences if $\gamma$ is rational.

I guess this does not change anything, except that it would be a truely sensational discovery. I think that there is no important conjecture based on the irrationaly of $\gamma$.

The number $e^{-\gamma}$ plays a role in some products of the primes, but the only consequence of the rationality of $\gamma$ would be that we could be sure that $e^{-\gamma}$ is transcendental.

So, besides that we could determine the status of more real numbers, I do not think the rationality of $\gamma$ would have any impact.