Elementary theorems with several proofs?

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

Basel problem has many interesting proofs.

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

Answer 4

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)

Answer 5

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).

Answer 6

"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.

Answer 7

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.