Just how good are polynomials at producing primes? Do there exist polynomials that produce primes for arbitrarily many consecutive inputs?
In this post, I’ll give a brief overview of what we expect to be able to prove, and show how interpolating polynomials can produce record-breaking prime-generators. (And then break a record, because why not?)
Classification theorems of Euler, Lagrange, and Legendre describe the sets of integers that can be written as the sum of 2, 3, and 4 squares. In the last two cases, it follows easily that the density of these sets are 5/6 and 1.
The question of density is not so simple in the case of two squares. In this post, we resolve using an unexpected tool — Dirichlet’s theorem on primes in arithmetic progressions.
This post concerns generalizations of Euclid’s proof of the infinitude of primes to number fields, where it can be viewed as a relationship between the size of the units group and the number of prime ideals in a ring.
a. w. walker 