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Can anyone suggest me an ingenious proof of the transcendence of $\pi$. I have seen Lindemann's proof but it appears intricate.

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up vote 10 down vote accepted

There is a very nice book, "Irrational Numbers" by Ivan Niven. Available in paperback from the M.A.A. Evidently he gives a proof in the M.A.A.'s American Mathematical Monthly, volume 46 (1939) pages 469-471. His comment in the notes for chapter 9 of the book has "Proofs of the transcendence of $e$ and $\pi$ are not so difficult as the proof of the more general Theorem 9.1" And his 9.1 is indeed Hermite-Lindemann-Weierstrass. $$ $$ See also Proof that pi is transcendental that doesn't use the infinitude of primes which had a specific emphasis.

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I have never seen a proof of the transcendence of $\pi$ which is anything but ingenious: that's rather the problem. I gather you want a proof which is as "transparent" as possible.

So let me mention a highly recommended book with that as its stated goal: Making Transcendence Transparent by Burger and Tubbs. For instance, this book received a rave review from (sometime MOer and full-time arithmetic geometer) Álvaro Lozano-Robledo on the MAA website:

I had a little trouble electronically searching for "transcendence of $\pi$" in this book. Rather, they view that result as a special case of the Lindemann-Weierstrass theorem, which seems like a good way to go.

Let me disclose that I have not read the book myself, but it is where I would start if I wanted to gain a better understanding of these issues.

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I heared there's still no known way to prove whether numbers like $ \pi + e $ are irrational. I take that as a sign that there isn't a transparent proof for $ \pi $ transcendent yet. – Zsbán Ambrus Dec 14 '11 at 13:32

An unfortunate thing is that proving the transcendence of $\pi$ has the same complexity as the Lindemann--Weierstrass theorem. (Proving the transcendence of $e$ is much easier.) I am quite surprised to see that the book [A.I. Galochkin, Yu.N. Nesterenko, and A.B. Shidlovskiĭ, Введение в теорию чисел [Introduction to number theory], 2nd edition, Moscow State Univ., 1995. 160 pp. ISBN: 5-211-03075-3. MR1367734 (96i:11001)] is not translated into English (I remember that there was an attempt to negotiate the translation about 10 years ago). I put here an extract from Yu. Nesterenko's lectures (in Russian!) on the Lindemann--Weierstrass theorem which are "cleaned", so I believe this is the simplest version of proof.

There are more exotic ways to state the transcendence of $\pi$, of course, these proofs are more involved. One example in this direction is [V.N. Sorokin, On the measure of transcendency of the number $\pi^2$, Sb. Math. 187 (1996), no. 12, 1819--1852] where the author shows the transcendence of $\pi^2$ (rather than $\pi$ itself) by constructing linear forms involving the numbers $$ \sum_{n_1\ge n_2\ge\dots\ge n_r\ge1}\frac1{n_1^2n_2^2\cdots n_r^2} $$ which are (now) known to be simple rational multiple of $\pi^{2r}$, $r=1,2,\dots$.

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Four pages is pretty good, and there is a very high proportion of formuas to text. H-L-W would be Theorem 4.28, and transcendence of $\pi$ is something 4.10, where the word something means either corollary or Russian style hors d'oeuvre – Will Jagy Aug 1 '10 at 3:05
Something 4.10 (Следствие 4.10) is indeed a corollary. Zakuski is also a corollary but this time of vypivka (drinking). :-) – Wadim Zudilin Aug 1 '10 at 3:42
Hmm... Следствие also means "criminal investigation", I don't get the relation to закуска (ед.ч.), which follows за выпивкой. BTW, "Предположим противное" (at the beginning of the proof of 4.28) means "let us assume the repugnant". I've always wanted to object: why not assume the pleasant ("предположим приятное")? I guess these proofs are written by pessimists. – Victor Protsak Aug 1 '10 at 4:50
You are definitely right, Victor! To have more fun from math (and, in particular, from linear algebra ;-) ) we have to be optimistic. Zakuski is just Will's famous word, and I keep respect to this. :-) – Wadim Zudilin Aug 1 '10 at 5:05
Originally I was going to say either Corollary or hamburger. Then I though I ought to stick with the familiar. A recent (2010) quote from the introduction of a book by Elif Batuman: "That's when I was sucked in, deeper than I ever expected. The title of the book is borrowed from Dostoevsky's weirdest novel, The Demons, formerly translated as The Possessed, which narrates the descent into madness of a circle of intellectuals in a remote Russian province: a situation analogous, in certain ways, to my own experiences in graduate school." – Will Jagy Aug 1 '10 at 5:28

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