Does anybody have a proof or know about the result by Borwein and Bailey that
$e^\sqrt{2}$ is irrational
and perhaps the motivation behind the problem and it's solution.
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@DietrichBurde - it did not pop up when I wrote the title -thanks. – Anthony Sep 27 '16 at 20:23
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1For the motivation see also here. – Dietrich Burde Sep 27 '16 at 20:23
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In particular I was interested in the sequence A123168 – Anthony Sep 27 '16 at 20:25
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I am going to edit my title and repost - I believe the sequence I have listed shows that $e^\sqrt{2}$ is irrational without using the Lindemann-Weirestrss Theorem – Anthony Sep 27 '16 at 20:31
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1This is also shown in the duplicate without Lindemann. – Dietrich Burde Sep 27 '16 at 20:32
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Using that sequence and that specific continued fraction? – Anthony Sep 27 '16 at 20:33
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I have to look ... But if not, you could pose a new question, and you should give the reference of OEIS in the new question. – Dietrich Burde Sep 27 '16 at 20:34
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This is not a "result by Borwein and Bailey". It follows from a result of Lindemann, which Borwein and Bailey were well aware of. – Robert Israel Sep 27 '16 at 20:42
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@robert israel can you elaborate? – Anthony Sep 27 '16 at 23:45
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Lindemann proved in 1882 that $e^\alpha$ is transcendental if $\alpha$ is a nonzero algebraic number. – Robert Israel Sep 28 '16 at 00:22
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I should have asked for an understanding of the continued fraction of $$\sqrt{2}\Bigg({e^\sqrt{2}-1 \above 1.5pt e^\sqrt{2}+1 }\Bigg)$$ and how this continued fraction proves the irrationality of $e^\sqrt{2}$ – Anthony Sep 28 '16 at 10:56