An elementary proof of the irrationality of Pi

I´ve been wondering for a while now if it exists and if so, how it looks like. By elementary I mean something like: completely (elementary) number theoretically. Because I know there are a few easy proofs that \pi is not rational, but all of the ones I’ve seen, make use of some calculus (like computing an integral or so). So, my question: Can anyone show me the irrationality of \pi with just the very basics of number theory or tell me  why it is/shouldn’t be possible? For example, a few years ago I tried to prove it using the the formula: \dfrac{\pi}{2} = \displaystyle \sum_{k=0}^\infty \dfrac{k!}{(2k + 1)!!} along with basic arguments which you may find in some proofs of the irrationality of e = \displaystyle \sum_{k=0}^\infty \dfrac{1}{k!}. Although (obviously) I wasn’t able to prove  the desired result, I did not find a reason why it shouldn’t be possible in principle. Even more, I had the gut feeling I was almost there. So I might give it another shot it the future, but hopefully that doesn’t stop you, my dearest reader, from helping me if you’ve got some info on this.

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5 Responses to “An elementary proof of the irrationality of Pi”

  1. My idea Says:

    Dear prof Math
    I want a question but I think I can wrong
    pi=4/(1+(1^2)/(2+(3^2)/(2+(5^2)/(2+(7^2)/(….)))))).?
    I read it in a book in past but I think my memory can wrong

  2. My idea Says:

    😉

  3. My idea Says:

    thank you,prof Math

  4. My idea Says:

    I like your blog very much,it’s great

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