TITLE: How many different proofs can a theorem have?
QUESTION [10 upvotes]: 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?

REPLY [6 votes]: 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