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From this list I came to know that it is hard to conclude $\pi+e$ is an irrational? Can somebody discuss with reference "Why this is hard ?"

Is it still an open problem ? If yes it will be helpful to any student what kind ideas already used but ultimately failed to conclude this.

  • 1
    According to mathworld, it's still an open problem: http://mathworld.wolfram.com/e.html2012-06-17
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    The same think is asked in (a part of) this question: http://math.stackexchange.com/questions/28243/is-there-a-proof-that-pi-times-e-is-irrational2012-06-17
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    I don't think this is precisely a duplicate of the other question, as this one asks for references and discussion about why previous techniques are insufficient to resolve the problem. (I've edited the title to match.) This can be more illuminating than a simple yes/no answer, which is what the previous question received.2012-06-17
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    After Rahul Narain's [edit](http://math.stackexchange.com/posts/159350/revisions) the title of the question corresponds to the body. So it seems that it is a different question - I apologize for being too quick in voting to close.2012-06-17
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    Why shouldn't it be hard?2012-06-17
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    @Qiaochu, I agree that the obvious answer to the stated question is "why not?", but negative results about what kinds of techniques *cannot possibly work* can still give much insight. For example, [there are several results regarding what classes of proofs are insufficiently powerful to resolve P vs. NP](http://en.wikipedia.org/wiki/P_versus_NP_problem#Results_about_difficulty_of_proof). I've upvoted this question because I guess I'm hoping to learn something similar here.2012-06-17
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    I think the expectation is that much more is true: $\pi$ and $e$ are algebraically independent. See http://mathoverflow.net/questions/33817/work-on-independence-of-pi-and-e.2013-06-04
  • 0
    If it were rational, it would be difficult2015-09-17

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