# How many different proofs can a theorem have?

I notice some problems has many different proofs, do all theorems have multiple proofs, is there some theorems which has only 1 way to prove it? $n$ ways? infinite?

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How would you define "one way"? You can switch the order a bit, do you then have a new proof? –  Jonas Teuwen May 4 '11 at 16:48
Before you can ask how many different proofs a theorem have, you have to decide what it means to be different. There is a discussion of this at Gowers' blog: gowers.wordpress.com/2007/10/04/… . –  Raeder May 4 '11 at 16:49
Have a look at this paper: mathdl.maa.org/images/upload_library/22/Ford/Wagon601-617.pdf. –  Yuval Filmus May 4 '11 at 17:06
The (many!) proofs of quadratic reciprocity are quite varied, I'm told. –  Ｊ. Ｍ. May 4 '11 at 17:16
The collection of all proofs of a theorem does not form a set. It at least forms a category with nontrivial isomorphisms, possibly an $\infty$-category... –  Qiaochu Yuan May 4 '11 at 18:43

I guess one way to interpret the question is how many proofs of a given theorem can be published, under the (somewhat dubious) assumption that two proofs have to be significantly different in order for both of them to get published. So I direct your attention to Murray Gerstenhaber's paper, The 152nd proof of the law of quadratic reciprocity, Amer. Math. Monthly 70 (1963) 397–398, MR0150097 (27 #100) (but I warn you that the title was somewhat tounge-in-cheek). See also Elisha Loomis, The Pythagorean Proposition, which has 365 proofs of that well-known result.

There have been many "proofs" of P = NP, and roughly an equal number of "proofs" of P $\ne$ NP. GJ Woeginger keeps track of them at http://www.win.tue.nl/~gwoegi/P-versus-NP.htm

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