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Q. What are the characteristics of theorems that seem to invite (or possess) several or even many distinct proofs?

What I have in mind are examples such as these:

The last is particularly striking to me, as it took some time for an accurate proof to emerge.1 I'm sure there are many other more modern examples; suggestions welcomed.

Is there some characteristic of these theorems that lend themselves to often rather distinct proofs? Or is it where within the network of mathematical connections these theorems reside? Or is it that these theorems are so useful that researchers keep inventing new proof approaches?

1Imre Lakatos, Proofs and Refutations. Wikipedia link.
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    $\begingroup$ Another prominent example is quadratic reciprocity. Does the fundamental theorem of algebra count? $\endgroup$
    – M.G.
    Commented Aug 18, 2020 at 0:20
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    $\begingroup$ The claim that two things are not equal (or variants of it) often has many fundamentally different proofs. This is because two things that are not equal are usually equal for many reasons - i.e. two integers are not equal because they are unequal mod $2$, unequal mod $3$, etc. Usually for an equality, there will be similarities between different proofs, and you can argue about whether they are really the same proof. $\endgroup$
    – Will Sawin
    Commented Aug 18, 2020 at 0:48
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    $\begingroup$ There are a plethora of proofs that there are $n^{n-2}$ trees on $n$ labelled points. I'd say an common feature of most such theorems is to be easy understand, hard enough to be challenging but not so hard as to be unapproachable. Then other criteria come into play, $\endgroup$
    – Aaron Meyerowitz
    Commented Aug 18, 2020 at 5:39
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    $\begingroup$ I don't have an answer, but two more examples: Robin Chapman collected $14$ proofs of $\zeta(2)=\pi^2/6$ at empslocal.ex.ac.uk/people/staff/rjchapma/etc/zeta2.pdf And $14$ is a good number – Stan Wagon won a prize for "Fourteen proofs of a result about tiling a rectangle." maa.org/programs/maa-awards/writing-awards/… $\endgroup$
    – Gerry Myerson
    Commented Aug 18, 2020 at 5:58
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    $\begingroup$ The example theorems all have one thing in common, which makes it far more likely that they will have many proofs: they are OLD. (Euler's formula is by far the newest of them, but even that is over 200 years old.) I would be surprised to learn of a >200-year-old theorem with fewer than, say, five proofs. $\endgroup$
    – HJRW
    Commented Aug 18, 2020 at 10:20

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I think two important characteristics are

  • The importance of the theorem itself
  • The extent of connections within the subject

These two are not, of course, mutually exclusive!

This book is a recent discussion of mathematical style which has 99 proofs. A really great read.

@BOOK{Ording2019,
  title =     {99 Variations on a Proof},
  author =    {Ording, P.},
  publisher = {Princeton University Press},
  address =   {Princeton and Oxford},
  year =      {2019},
  isbn =      {9780691158839},
}

and this book collects together proofs of $\sqrt{2}$ is irrational.

@BOOK{Duchene2010,
  author =       {Duch\^{e}ne, L. and LeBlanc, A.},
  title =        {Rationnel mon Q: 65 exercices de styles},
  publisher =    {Hermann},
  year =         {2010},
}
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    $\begingroup$ "Rationnel mon Q" ^^ $\endgroup$
    – Olivier Bégassat
    Commented May 5, 2022 at 13:25
  • $\begingroup$ Indeed, the title of the second book is a pun that does not excel in taste; its French pronunciation can be translated as "rational my ass" (the letter "Q" and the French word "cul" have the same French pronunciation, with "cul" meaning "ass"). $\endgroup$
    – Alex M.
    Commented May 8, 2022 at 7:04
  • $\begingroup$ Does this book credit Tom Apostol with a proof he published in 2000 and that I presented in a classroom some years before that after I learned it from a book published in about 1960? $\endgroup$
    – Michael Hardy
    Commented Sep 10, 2023 at 5:42

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