The Infinitude of Primes (IP) is a prominent example in the discussion on simplicity and purity of proofs. As a simple theorem of number theory, IP has been proved repeatedly by such famous mathematicians as Goldbach, Euler, Lebesgue, Kronecker and Erdös incorporating many different technics from arithmetic, algebraic number theory, analysis or topology.

Here we list 9 references for different proofs of IP only after the 1950’s.

By investigating these proofs using technics from proof theory, we aim to get a clearer picture of the phenomenon of simplicity and purity of proofs. In an ongoing series of papers we try to accomplish this goal by establishing an axiomatic treatment of several proofs and look for proof theoretic means of comparing them:

• Part I: Three Euclidian Proofs of IP