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Is the number $$\int_1^2 \frac{e^x} x \, dx=Ei(2)-Ei(1)$$ transcendental ?

The first few digits are

$$3.059116539645953407912984195895401006500992980687334462880866822688\cdots$$

Working with lindep and algdep-command with PARI/GP I did not find an indicitation that the given number is algebraic. But can it be proven that it is transcendental ?

I tried to apply the known powerful theorems (Lindemann-Weierstrass and Baker ), but without success.

If the number cannot be verified, is at least the status of $Ei(1)$ and $Ei(2)$ known ?

Here

https://en.wikipedia.org/wiki/Exponential_integral

the definition of $Ei(x)$

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  • $\begingroup$ By a probability criterion it must be :p $\endgroup$ – Masacroso Oct 2 '16 at 16:36
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    $\begingroup$ the first "few" digits, you say ... $\endgroup$ – Jean Marie Oct 2 '16 at 16:38
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    $\begingroup$ The number can also be written as $$\ln(2)+\sum_{k=1}^\infty \frac{2^k-1}{k\cdot k!}$$ $\endgroup$ – Peter Oct 2 '16 at 16:39
  • $\begingroup$ @JeanMarie Less than $100$ digits ... $\endgroup$ – Peter Oct 2 '16 at 16:47

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