What is the theorem that has the most proofs?

Solution 1:

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

Solution 2:

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

Solution 3:

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"