# What is the theorem that has the most proofs?

Classical theorems like the irrationality of $\sqrt{2}$ or the infinitude of the primes have lots of proofs. But one theorem in particular, which I studied years ago in an introductory course of Number Theory, called the Quadratic Reciprocity Law, has tons of proofs. Gauss himself provided some of them.

And the question is: Is here a theorem that has more proofs than the Quadratic Reciprocity Law? If you know other theorems that have lots of different proofs, please list below.

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Maybe we can divide the theorems by area.

In Euclidean Geometry, the Pythagorean Theorem won the game.

In Classical Number Theory, the Quadratic Reciprocity Law won the game (I think).

Now, what about Calculus? Algebraic number theory? Real Analysis? Commutative Algebra? Topology? Differential Geometry? Differential Equations? Probability? And so on.

• As simple facts may turn out to have many proofs (a recently bumped-up question is a good example), I suppose you mean named theorems. If so, I think Cauchy-Schwarz inequality and A.M. $\ge$ G.M. are good contenders in the inequalities category. For linear algebra, I'll bet on Cayley-Hamilton theorem. The fundamental theorem of algebra also has many proofs, but I'm not sure what category it belongs to. Apr 4 '17 at 13:52
• I imagine there are various theories for which you can construct an infinite number of different proofs. Apr 4 '17 at 14:10
• @Filburt: We'll need a better definition of "proof" here, and also what it means for proofs to be identical. Note that in the abstract, you cannot prove for any two proofs whether they're identical, as that is a task of unbounded complexity. Apr 4 '17 at 15:02
• Your last comment is false or at best meaningless. If you know logic you will realize that it's extremely difficult to define proof equivalence that can distinguish between 'different proofs' in the same way that professional mathematicians would. And mathematicians don't even always agree... Apr 4 '17 at 16:15
• Well countless of logicians (and myself at one point) have pondered this intuitive question to no satisfactory conclusion. So you're not alone, but it really doesn't have a good answer, and is nearly totally opinion-based. For instance I don't agree that there are hundreds of proofs of the Pythagorean theorem, though I didn't downvote your question nor any of the answers. Apr 4 '17 at 17:32

Well, there was a book with 367 proofs of the Pythagorean Theorem published. I'm sure there's more.

• Well, that is insane. =) Apr 4 '17 at 1:53
• Maybe there will be a time when the number of proofs for the Riemann Hypothesis exceeeds this number. :-)
– Jens
Apr 4 '17 at 8:54

Proofs of Euler's polyhedral formula in The Geometry Junkyard:

Proof 1: Interdigitating Trees
Proof 2: Induction on Faces
Proof 3: Induction on Vertices
Proof 4: Induction on Edges
Proof 5: Divide and Conquer
Proof 6: Electrical Charge
Proof 7: Dual Electrical Charge
Proof 8: Sum of Angles
Proof 9: Spherical Angles
Proof 10: Pick's Theorem
Proof 11: Ear Decomposition
Proof 12: Shelling
Proof 13: Triangle Removal
Proof 14: Noah's Ark
Proof 15: Binary Homology
Proof 16: Binary Space Partition
Proof 17: Valuations
Proof 18: Hyperplane Arrangements
Proof 19: Integer-Point Enumeration
Proof 20: Euler tours
• Are any of these proofs correct? The formula invites an excessive variety of counterexamples. Apr 4 '17 at 17:38
• @djechlin, the Euler formula is true under the correct hypothesis. See Proofs and refutations by Imre Lakatos. Apr 4 '17 at 17:41
• I looked through the book. It seems to be about how a huge number of hypotheses is needed to address all of the counterexamples. Is there a single hypothesis that works that's mentioned somewhere in the book? Apr 4 '17 at 17:49
• @djechlin, simply connected polyhedron with simply connected faces. Convex is sufficient. Apr 4 '17 at 18:02
• Does the case of a small cube sitting on top of, centered (or at the back), on a large cube, fail polyhedral? Apr 4 '17 at 18:05

There are also a lot of ways to prove that there are infinitely many primes.

See for example: Euclid's theorem on the infinitude of primes: A hisorical survey of its proofs on arXiv, by Romeo Meštrović. Of course, a lot of them look like each other, but there are several different approaches as well.

"In this article, we provide a comprehensive historical survey of 169 different proofs of famous Euclid’s theorem on the infinitude of prime numbers"

There are actually a surprisingly large number of ways to prove the Fundamental Theorem of Algebra, ranging from real analysis to complex analysis to Galois theory to Riemannian Geometry. At least one example of each of these categories is listed here. Not sure what category you want to count that theorem as, but if it's a sufficently sparse category it probably wins.

For logic, Godel's Completeness Theorem has a bunch of proofs too. I know one that uses Tychonoff's Theorem to prove the Compactness Theorem first, the original one Godel gave, Henkin's famous proof, and one that directly constructs a model. I bet I could (somewhat circuitously) prove it using graph theory and Zorn's Lemma too actually. Godel's Incompleteness Theorem also has several proofs. Not sure which of the two has more proofs but I'm leaning towards the Completeness Theorem.

The theorem that $\sum\frac{1}{n^2}=\frac{\pi^2}{6}$ has a lot of proofs, ranging from probability theory to complex analysis to real analysis to Fourier analysis to number theory. There's even a proof that is experimentally verifiable, insofar as there is a particular experiment that can be proven to return "yes" instances with probability $\frac{1}{\zeta(2)}$ and so you can go out and do the experiment and find out that it happens with probability $\frac{6}{\pi^2}$. This experimental proof generalizes to the following statement:

For any number $n$, pick $k$ positive integers less than $n$ uniformly at random. Let $p(n,k)$ be the probability that all $k$ numbers are relatively prime. Then $$\lim_{n\to\infty}p(n,k)=\frac{1}{\zeta(k)}$$ With a handful of dice, you can carry this experiment out yourself! See here for details.

• Can you give a reference to the 'experiment' proof ? Thanks! Apr 4 '17 at 19:17
• @SandeepSilwal Fix some large number $n$ and selected $k$ positive integers less than $n$ uniformly at random. Call the probability that they are relatively prime $p(n,k)$. Then $\lim_{n\to\infty}p(n,k)=\zeta(k)^{-1}$. With a few ten-sided dice, you can carry this experiment out yourself! See here: qchu.wordpress.com/2010/11/09/… Apr 4 '17 at 19:31
• In my first year at university at least 4 different lecturers proved the fundamental theorem of algebra using the subject they happened to be teaching. One used heat diffusion (true story). So I think that must win for breadth of proofs, even if others have more listed. Apr 4 '17 at 19:35
• You can also prove the fundemental theorem of algebra using functional analysis, the spectrum of a compact operator being nonempty.
– Alan
Apr 23 '17 at 22:44

It must be Pythagoras's theorem. There's hundreds of proofs (famously, a whole book of them). Cut-the-Knot has a few of them...

• I changed the question to make it more interesting. Apr 4 '17 at 2:01

In algebraic number theory, there are a lot of ways of getting to the main results of class field theory.

Here are what I think of as the main results. Of course there are many auxiliary results, which appear either as corollaries of theorems (1) and (2) or are part of their proof, such as Kronecker-Weber theorem, Hilbert class field theorem, existence of global and local Artin maps, determination of congruence conditions on the splitting of primes, and the determination of lower reciprocity laws.

1 . (Global Class Field Theory): Let $$K$$ be a global field. The map $$L \mapsto K^{\ast}N_{L/K}(\mathbb{I}_L)$$ gives an order preserving bijection between finite abelian extensions of $$K$$, and open subgroups of $$\mathbb{I}_K$$ containing $$K^{\ast}$$, where $$\mathbb{I}_K$$ is the group of ideles of $$K$$. Under this mapping, $$[L : K] = [\mathbb{I}_K : K^{\ast}N_{L/K}(\mathbb{I}_L)]$$.

2 . (Local Class Field Theory): Let $$F$$ be a local field. The map $$E \mapsto N_{E/F}(E^{\ast})$$ gives an order preserving bijection between finite abelian extensions of $$F$$ and open subgroups of $$F^{\ast}$$. Under this mapping, $$[E : F] = [F^{\ast} : N_{E/F}(E^{\ast})]$$. Moreover, $$[\mathcal O_F^{\ast} : N_{E/F}(\mathcal O_E^{\ast})] = e(E/F)$$

Methods of proof (I'm not totally familiar with all of these, so hopefully someone will correct me on any errors):

1 . Use analytic methods to prove that $$[\mathbb{I}_K : K^{\ast}N_{L/K}(\mathbb{I}_L)] \leq [L : K]$$, group cohomology to prove that $$[E : F] = [F^{\ast} : N_{E/F}(E)^{\ast}]$$ for cyclic local extensions $$E/F$$, more group cohomology to prove that $$[\mathbb{I}_K : K^{\ast}N_{L/K}(\mathbb{I}_L)] = [L : K]$$ for cyclic global extensions $$L/K$$. Simultaneously prove the existence of a well defined global Artin map $$\mathbb{I}_K \rightarrow \textrm{Gal}(L/K)$$ with kernel $$K^{\ast}N_{L/K}(\mathbb{I}_L)$$ and complete the proof that $$[\mathbb{I}_K : K^{\ast}N_{L/K}(\mathbb{I}_L)] = [L : K]$$ for all $$L/K$$ finite abelian. Use the global Artin map to prove that open subgroups of $$\mathbb{I}_K$$ containing $$K^{\ast}$$ correspond to finite abelian extensions of $$K$$, thereby completing theorem (1), and at the same time use these results to define the local Artin map and complete theorem (2).

2 . The same as the first approach, except skip the analytic methods; one can give a cohomological proof instead.

3 . Either of the first two approaches, but instead of using the ideles, use generalized ideal class groups and the formalism of cycles. This has the advantage of transitioning naturally to the discussion of congruence conditions on the splitting of primes, but is more clumsy for the discussion of infinite abelian extensions.

4 . Local class field theory first, using Lubin-Tate formal groups to define the local Artin map. One can then define the global Artin map on the ideles from the local ones, and then obtain the results for global class field theory.

5 . Again local class field theory first, using group cohomology to define the local Artin map and prove the main results of local class field theory.

6 . Class field theory using central simple algebras. This is an approach I know nothing about.

7 . (Work in progress) Obtain theorems (1) and (2) as a result of Langlands functoriality.

References: for 1, 2, 3, Lang's Algebraic Number Theory and Artin-Tate Class Field Theory. 4, Milne's notes on class field theory. 5, Cassels and Frohlich, Algebraic Number Theory. 6, Milne again.