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Let $u$ be the solution of the equation $$e^x \log(x)=1$$

Is $u$ rational, irrational algebraic or transcendental?

$u$ seems to be transcendental, but I cannot prove it.

Perhaps, someone has an idea.

Peter
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    I apologize for the format, but this website uses an annoying filter for "quality standards". It seems that a question is not allowed to be short and direct. Do not ask me why ... – Peter Oct 11 '13 at 08:18
  • The numerical value of u is 1.30979 95858 04150 47767 , rounded to 20 digits. – Peter Oct 11 '13 at 08:19
  • Tabulated to 99 digits at http://oeis.org/A201942/b201942.txt (but with no helpful information) – Gerry Myerson Oct 11 '13 at 08:40
  • I calculated u to 100 000 digits with PARI. Looking at the continued fraction, brought me to the conjecture, that u is transcendental. – Peter Oct 11 '13 at 08:55
  • The theorems of gelfond-schneider and lindemann-weierstrass do not seem to help. – Peter Oct 11 '13 at 08:56
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    What is it about the continued fraction that you take to distinguish transcendentals from algebraic irrationals? Next to nothing is known about about the continued fraction of, say, $\root3\of2$. – Gerry Myerson Oct 11 '13 at 08:58
  • The vector of the convergents has high numbers. Of course, this does not proof anything. – Peter Oct 11 '13 at 09:03
  • How high? Higher than for $\root3\of5$? – Gerry Myerson Oct 11 '13 at 09:04

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