Other constants such as $\pi$, $e$, $\phi$, $\zeta(3)$ etc, had been proof to be of irrational constants.

There are many series, infinite products and integrals that representing Euler's constant and yet it is still an open problem of its irrationality mystery.

What makes Euler's constant so hard to prove that it is an irrational or not as a constant?

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    $\begingroup$ See here : mathoverflow.net/questions/129364/… $\endgroup$ – Rohan Jan 30 '17 at 8:31
  • $\begingroup$ Similar to Fermat last theorem: what takes it so long to prove it(350 years later ). Maybe at this moment they haven't develop the mathematical tools to tackle such simple maths problem. $\endgroup$ – user348832 Jan 30 '17 at 8:36
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    $\begingroup$ There are lots of conjectures in this subject, but rather few theorems, and nothing remotely approaching a general method applicable to all the "obviously" irrational numbers. $\endgroup$ – Robert Israel Jan 30 '17 at 9:03

Other constants such as $\pi$, $e$, $\phi$, $\zeta(3)$ etc, had been proof to be of irrational constants.

Perhaps. But, in case you haven't noticed it by now, those proofs of irrationality are not one and the same. In other words, there is no catch-all method for proving that something is irrational in general. Various methods do exist for various situations $($such as the Gelfond-Schneider theorem, for instance$)$, but they do not cover all possible cases. Indeed, they don't even cover a majority of cases, but only some countable subset, whereas irrationals $($ more specifically, transcendentals $)$ are uncountable. Hope this helps.

  • $\begingroup$ Thank you @Lucian, interesting and concise information. $\endgroup$ – gymbvghjkgkjkhgfkl Jan 30 '17 at 20:54
  • $\begingroup$ But we can only hope to prove anything about numbers that have a definite description, and the number of descriptions is countable.. $\endgroup$ – Henning Makholm Mar 7 '17 at 16:06
  • $\begingroup$ @HenningMakholm: Even if the number of descriptions were finite (!), it is enough for a single element of such a description to vary over an uncountable set, in order for the number of terms classified into a finite number of descriptions to be uncountable; e.g., all positive numbers can be written in terms of the Taylor series for the exponential function, whose argument varies over the reals. $\endgroup$ – Lucian Mar 7 '17 at 20:20

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