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There are many numbers which are not able to be classified as being rational, algebraic irrational, or transcendental. Is there an explicit number which is known to be irrational but not known to be either algebraic or transcendental?

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  • $\begingroup$ I would think numbers like $.121121112\dots$ would do the job, no? Liouville doesn't obviously give us transcendence (at least it isn't obvious to me). $\endgroup$ – lulu Apr 18 '16 at 18:07
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    $\begingroup$ $e+\pi$ or $e\cdot\pi$. $\endgroup$ – student forever Apr 18 '16 at 18:07
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    $\begingroup$ @studentforever: I don't know that either of those numbers has the indicated property. (Of course at most one of them is rational, but I can't see how that helps here.) $\endgroup$ – Charles Apr 18 '16 at 18:09
  • $\begingroup$ @lulu I believe that number is known to be transcendental. It's the sum of a rational number ($\frac19$) and a theta-value at a rational argument that I'm pretty sure is known not to be algebraic. $\endgroup$ – Steven Stadnicki Apr 18 '16 at 18:20
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    $\begingroup$ Erdős proved that the Erdős–Borwein constant $\sum_{n = 1}^{\infty} \frac{1}{2^n - 1} = 1.60669\!\ldots$ is irrational, and to my knowledge whether it's algebraic remains open; the comments below this old answer suggest that's the case: math.stackexchange.com/a/266638/155629 . en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Borwein_constant $\endgroup$ – Travis Apr 18 '16 at 18:26
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Maybe the best-known example is Apery's constant, $$\zeta(3) = \sum_{n = 1}^{\infty} \frac{1}{n^3} = 1.20205\!\ldots ,$$ which Apery proved was irrational a few decades ago; this result is known as Apery's Theorem.

By contrast, $\zeta(2) = \sum_{n = 1}^{\infty} \frac{1}{n^2}$ has value $\frac{\pi^2}{6}$, which is transcendental because $\pi$ is.

Apéry, Roger (1979), Irrationalité de $\zeta(2)$ et $\zeta(3)$, Astérisque (61), 11–13.

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    $\begingroup$ And $\zeta(2n+1)$ in general for $n\geq 1$ is not known either, I think. $\endgroup$ – Ravi Apr 18 '16 at 18:17
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    $\begingroup$ @Jake I think that's true, but I don't think it's known for any odd $m > 3$ whether $\zeta(m)$ is even rational. It is known, perhaps a little oddly, that at least one of the first four such numbers is irrational. (See Zudilin, Wadim (2001), One of the numbers $\zeta(5)$, $\zeta(7)$, $\zeta(9)$, $\zeta(11)$ is irrational, Russ. Math. Surv. 56 (4), 774–776.) $\endgroup$ – Travis Apr 18 '16 at 18:20
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    $\begingroup$ Ah yes. I was being hasty. Thank you :) $\endgroup$ – Ravi Apr 18 '16 at 18:44
  • $\begingroup$ Wow, I thought I knew the Greek symbols by sight, but I had to look up that weird salamander-looking thing to find out it's a lower-case Zeta. $\endgroup$ – Mason Wheeler Apr 18 '16 at 19:44
  • $\begingroup$ @MasonWheeler: in maths undergrad terms that's "other squiggle", with "squiggle" being $\xi$. $\endgroup$ – Steve Jessop Apr 19 '16 at 7:51
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The most famous have been answered. Let us be a little less constructive. At least one of $\zeta(5)$, $\zeta(7)$, $\zeta(9)$, $\zeta(11)$ is irrational, a result due to V. V. Zudilin, Communications of Moscow Mathematical Society (2001), and their true nature (algebraic and transcendental) seems unknown at the present time. This result improves the irrationality of one of the nine numbers $\zeta(5)$, $\zeta(7)$, $\ldots$ $\zeta(21)$.

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  • $\begingroup$ Is it known that these numbers are all irrational? $\endgroup$ – Charles Apr 18 '16 at 18:22
  • $\begingroup$ @Charles not by me, but at least of them (explicitely) is irrational, and unknown to be algebraic or transcendental. This is why I called my answer non constructive $\endgroup$ – Laurent Duval Apr 18 '16 at 18:35
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    $\begingroup$ Fair enough. :) $\endgroup$ – Charles Apr 18 '16 at 18:38
  • $\begingroup$ @Charles glad you liked it, pfiou... $\endgroup$ – Laurent Duval Apr 18 '16 at 21:07

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