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.
The prime number theorem (PNT) was not proven until 1896, but a weaker form (up to constants) was established decades earlier. The earliest proof was due to Chebyshev in 1852, and his work inspired others to take up the mantle and inch towards what they thought would be a proof of the PNT. Here, we show the strength of the Chebyshev method and ask whether it had the power to prove the PNT after all.
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 