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A while back I came across a result about non-simple continued fractions that allows proving that some numbers are irrational. The result in modern terminology is:

If, in the continued fraction $$\cfrac{b_1}{a_1 + \cfrac{b_2}{a_2 + \cfrac{b_3}{a_3 + \dots}}}$$ the values $a_{i}, b_{i}$ are all positive integers, and if we have $a_i \geq b_i$ for all $i$ greater than some $n$, then the value of the continued fraction is irrational.

I have checked that this result (which has been known for well over a century) can be used to prove the irrationality of $\pi$ and (non-zero) rational powers of $e$ (see here).

My question is this:

Does anyone know of any other uses of this old theorem for proving other constants are irrational? I am not naive enough to believe that mathematicians might have overlooked a neat proof of the irrationality of Euler's $\gamma$ or $\zeta(2n+1)$, but have searched in vain for any other uses, and not been able to find any by my own investigations.

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This is not an answer, just a comment from A few remarks on ζ(3) - Yu. V. Nesterenko

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  • $\begingroup$ Many thanks. So ... if we could just modify that continued fraction to make the old theorem apply ... :) $\endgroup$
    – Old John
    Nov 5, 2012 at 15:42

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