Elementary theorems with several proofs?

Every year my student's math club organizes a "proof marathon", where we present multiple proofs for a single theorem. For instance, last edition we did the AM-GM inequality with geometric, algebraic, analytic... proofs, and even one "proof" based on the laws of thermodynamics.

A list of topics we already did:

• Euclid's theorem (the infinitude of the primes)
• The Pythagorean theorem
• The divergence of the harmonic series
• The AM-GM inequality

For all of these topics, we were able to find at least 10 short proofs with lots of variety.

Some other subjects I'm considering:

• The irrationality of $\sqrt{2}$
• Euler's polyhedra formula
• Fermat's little theorem

Which other theorems or results lend themselves to such a proof marathon? We're looking for easy-to-understand theorems with several short and variated proofs.

Several proofs that the group $(\mathbf Z/(p))^\times$ is cyclic: http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/cyclicmodp.pdf.

Several proofs of the evaluation of the Gaussian integral from probability theory: http://www.math.uconn.edu/~kconrad/blurbs/analysis/gaussianintegral.pdf.

• I really like the Third Proof: Bounding with the Maximal Order. It also seems to work for any multiplicative group (group of units) of a field. Aug 13, 2014 at 3:10

Basel problem has many interesting proofs.

Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$

I think I have seen many fundamentally different proofs of the Fundamental Theorem of Algebra, some of them easy to understand. And now I see there is a math.stack question on this here.

Uncountability of $\mathbb R$ (and countability of $\mathbb Q$, $\mathbb Q^2$, $\mathbb Q(\sqrt 2)$...)

The fact that Euler characteristic of a triangulated object does not depend on the triangulation.

More sophisticated theorems:

Brower fixed point theorem

Theorem of invariance of the domain (or show that $\mathbb R^2$ is not homeomorphic to $\mathbb R^3$)

Let me add also the Jordan curve theorem (if one wants a simplyfied version one can restricto to polygonal curves)

• Is there a simple proof of the domain invariance without using homology theory? Jul 20, 2014 at 13:02
• It depends how what you classify "simple". There are profs by standard "analysis" means. But you need some tools like approximation of contiunuous functions by polynomals etc... In te view of the OP, one could consider weak version of the statement, allowing map to be smooth or piece wise linear, depending on the level of the working group. Jul 20, 2014 at 13:21
• b.t.w. Here you find a nine account of analytic aspect of the theorem of invariance of domain. terrytao.wordpress.com/2011/06/13/… Jul 20, 2014 at 13:21

Should there be anyone else interested, I've also stumbled upon sixteen proofs of the isoperimetric problem (link) and fourteen of a generalization of De Bruijn's packing theorem (link).

"Whenever a rectangle is tiled by rectangles each of which has at least one integer side, then the tiled rectangle also must have at least one integer side."

These Fourteen Proofs of a Result about Tiling a Rectangle are not all essentially different, but they show a wide variety. One uses a double integral, another the existence of infinitely many prime numbers.

• Also, different proofs generalize in different directions, and this is explored nicely in the paper. Aug 12, 2021 at 2:16
1. the existence of transcendental number
2. uncountability of R, 1) Contor's diagonal; 2) Contors's nested intervals; 3) forcing/Rasiowa-Sikorski lemma (we add a new real to a countable model of zfc); 4) Baire Category Theorem; 5) categoricity of countable dense order...
3. Arrow's theorem:If there are at least three distinct social states and a finite number of individuals, then no social welfare function can satisfy U (unrestricted domain), P (Pareto principle): If everyone prefers any x to any y, then x is socially preferred to y., I (Independence of irrelevant alternatives), D (Non-dictatorship)

Arrow's theorem has dozens of proofs, as Sen said, "Providing short proofs of Arrow's theorem is something of a recurrent exercise in social choice theory."(https://econweb.ucsd.edu/~rstarr/113Winter2012/Sen's%20ARRO-COL%2009A.pdf , p.3, footnote 5)

Arrow's own approach is to show that U, P, D and I will contradict finity.

Other approaches is to show that P, I, D contradict U or U, I and D contradict P, P, I, D, U will lead to a infinity of voters, and so on.