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Theorem with the most proofs

  1. Mar 5, 2015 #1
    I wonder which theorem has the most proofs, or has been proven in the most ways? I know of Loomis' The Pythagorean Proposition which came out decades ago & contains 370 proofs & more, & the proofs are even catalogued into four types (algebraic, geometric, etc). So that makes me think the Pythagorean theorem is the one. What about the infinitude of primes though? Or maybe there's a theorem I haven't thought of?
  2. jcsd
  3. Mar 5, 2015 #2


    Staff: Mentor

  4. Mar 6, 2015 #3
    Yeah, so I'm going to be annoying and ask what a "different" proof is. Because it seems very easy to give infinitely many proofs of theorems by adjusting some details. So I think it is interesting to think about what makes a proof essentially different from another.

    So to actually answer your question, there's the fundamental theorem of algebra which has many proofs too, but I think it's going to hard to beat the Pythagorean theorem.
  5. Mar 6, 2015 #4
    That hadn't occurred to me. You don't mean simply verifying the Pythagorean theorem for each Pythagorean triple, which would give you a countably-infinite number of 'different' proofs do you or is it something else?

    re: fundamental theorem of algebra I vaguely remember flipping through this book a long time ago. it has a bunch of different proofs of it:

    edit: added 'remember'
    Last edited: Mar 6, 2015
  6. Mar 6, 2015 #5


    Staff: Mentor

    No, that's not what micromass meant. He was talking about the details of the proof, not verifying that the proof worked by testing an infinite number of examples.
  7. Mar 6, 2015 #6
    I still don't think I get it. Take two squares, a big one & a small one, where the smaller one is inside the big one & tilted so its four corners touch the sides of the big square. So the side length of the big square is a+b & the side length of the smaller one is c. If you calculate the area of the bigger square, first by multiplying out (a+b)2 = a2 + 2ab + b2 and then by adding together the areas of the four right triangles whose side lengths are a, b & c & the area of the inner square you get 4*(ab/2) + c2 = 2ab + c2. Equate both sides because they're two ways of calculating the area of the big square, cancel the 2ab & you get the theorem.

    Is that one proof that could be adjusted to get another one? How would I do that here? What am I missing?
  8. Mar 6, 2015 #7
    OK, so first we need to talk about what a proof is. Pure formally, you have a list of axioms and inference rules. Then a proof is a list of statements, each is either an axiom or follows directly from a previous statement on the list and the use of an inference rule.

    So in principle, every proof should be such a formal proof (which is very difficult to read) or it must be obvious that you can restate one as such. Here is an example of a formal proof: http://us.metamath.org/mpegif/zorn2.html
    So the above proof you give must first be restated as a formal proof.
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