Chapter 0 An Infinity of Primes Truth is on a curve whose asymptote our spirit follows eternally. eo Errera We begin with one of the greatest theorems in mathematics. Theorem 0.1. There is an infinite number of primes. Our proof is not that of Euclid and not better than the proof of Euclid, but it illustrates the theme of this work: looking at the mathematical world through an asymptotic lens. We begin as Euclid did. Assume Theorem 0.1 is false. Let p1,...,pr be a listing of all of the primes. For any nonnegative integer s, the unique factorization theorem states that there is a unique way to express (0.1) s = p1 α1 p2 α2 · pr αr , where α1,α2,...,αr are nonnegative integers. We turn this into an encoding of the nonnegative integers by creating a map Ψ, (0.2) Ψ(s) = (α1,...,αr). Let n 2 be arbitrary, though in the application below it shall be large. The integers s, 1 s n, are each mapped by Ψ to a vector 1
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