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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?

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    $\begingroup$ So your question is: "If it is rational, what would it imply mathematically?" correct? $\endgroup$ – Surb Feb 1 at 21:53
  • $\begingroup$ I think the OP is asking if it is possible it is rational, then why are we putting so much effort into trying to prove it is irrational. The reason is that heuristically, there is almost no chance it is rational, we would assign it a probability of essentially 0. $\endgroup$ – Matthew Liu Feb 1 at 22:04
  • $\begingroup$ @MatthewLiu, you understood my question perfectly right. On the other hand, I am indeed interested in how mathematics is affected if it is rational. $\endgroup$ – Zuriel Feb 1 at 22:18
  • $\begingroup$ @MatthewLiu, how about hundreds of years ago, people were conjecturing that Fermat numbers are all primes? $\endgroup$ – Zuriel Feb 1 at 22:23
  • $\begingroup$ Yes, it is possible (in the sense that we cannot prove otherwise), albeit very unlikely. See this MO thread for examples of a similar type. But the numerator and denominator would have to be very large, and it certainly would be surprising to see! $\endgroup$ – Jair Taylor Feb 1 at 22:27
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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.

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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.

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  • $\begingroup$ I wouldn't call it wasted. Presumably a lot of this research would have given insight into stuff like the harmonic series and the nature of irrational numbers even though the end result was negative. $\endgroup$ – Arthur Feb 1 at 22:14
  • $\begingroup$ You are right. I've edited my final sentence. $\endgroup$ – José Carlos Santos Feb 1 at 22:19
  • $\begingroup$ A similar situation arose with Fermat's last theorem. Though the theorem was known to be true, the proof was incredibly difficult. In fact it came as a side consequence of a much larger result. I don't think that the mathematical community sees that as wasted effort, but to the layman there is no impact. $\endgroup$ – Yves Daoust Feb 19 at 14:45
  • $\begingroup$ Fermat's last theorem was not "known" to be true BEFORE it was proven. Of course, the "odds" were clearly on the side of the truth. This is also true in the case of the irrationality of $\gamma$, but the situation is different because we still have no proof, and can therefore not rule the contrary out. By the way, even if Fermat's last theorem would have turned out to be false, this would not at all mean that mathematicians would have wasted their time. $\endgroup$ – Peter Feb 20 at 17:12
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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.

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  • $\begingroup$ An unlikely interesting possibility: $q$ is too large to be handled by any supercomputer, or whatever technology even in the next 1000 years. What can human beings do about this then? $\endgroup$ – Zuriel Feb 2 at 17:25
  • $\begingroup$ In this (unlikely) case it is hard to imagine that we ever would find out that $\gamma$ is rational. $\endgroup$ – Peter Feb 20 at 17:01

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