[Math] Theorems with many distinct proofs

Question

I was told that whenever one learns a new technique, it is a good idea to see if one can prove a well-known theorem using the new technique as an exercise. I am hoping to build a list of such theorems to test my technique.

My Question. What are some theorems with a.) many different proofs, or b.) proofs using strikingly different techniques from mathematics?

Theorems with multiple proofs:

• Fundamental theorem of algebra
• Fundamental theorem of arithmetic
• Existence of Jordan normal form. (Standard minimal polynomial proof, PID proof, Terry Tao's proof, proof using complex analysis)
• (Related to prev.) Cayley-Hamilton theorem
• Spectral theorem
• Quadratic reciprocity law
• Pólya's recurrence theorem
• Basel problem
• Stirling's formula

(Arguably:)

• Uniform boundedness principle. (Pf 1. Baire category, Pf 2. Gliding hump)
• Brouwer fixed point theorem (although all the proofs I know of boil down to showing that the ball is not homeomorphic to the sphere)

I think the Pythagorean theorem also satisfies my description, but it is a bit too elementary.

Clarification 1: In order to prevent this question from being "what are some theorems that have higher level generalizations," a trivial specialization of a harder theorem (e.g. Hilbert space Pythagorean theorem implying Euclidean Pythagorean theorem) will not be considered a new proof, for the purposes of this question.

Clarification 2: This question differs from this similar stackexchange question in the following sense. The stackexchange post asked for a.) very elementary theorems (lower level undergrad), and b.) short proofs. My question asks for theorems from all levels of mathematics (up to say third year graduate level), which are (preferably) central to the theory.

Answer 1

Terry Tao has called Szemerédi's theorem the "Rosetta stone" of mathematics, and for a good reason: this result has gathered many different proofs using tools from many different areas of mathematics. Wikipedia has a brief overview, but for completeness here is a list (feel free to edit to add ones I missed):

• Szemerédi's original, purely combinatorial proof (based on his regularity lemma and prior proof of the 4AP case)
• Furstenberg's proof using ergodic theory,
• Gowers's proof using Fourier-analytic methods (greatly refining Roth's method for 3APs)
• Another proof by Gowers, using hypergraph regularity results,
• Polymath1 project has resulted in an elementary proof of density Hales-Jewett, of which Szemerédi's theorem is an easy consequence.

They also have countless variants, e.g. this simplified proof of the density Hales-Jewett by Dodos, Kanellopoulos, Tyros.