Using Groups to Factor Integers

Most factorization algorithms in use today fit in one of two camps: sieve-based methods based on congruences of squares, and algorithms based on decompositions of algebraic groups. In this article, we trace the common thread connecting the latter.

The Forgotten Method of Prosthaphaeresis

In this post, I discuss the method of prosthaphaeresis, a proto-logarithm that enabled celestial navigation for the quarter-century predating the introduction of Napier’s logarithm.

Integrating Secant

The integral formula for secant is often introduced in a very artificial way. In this post, I look at the history of this integral and give several derivations.

Large Prime Factors of a Quadratic Polynomial

One of Landau’s four problems from 1912 concerns the infinitude of primes in the values of a certain quadratic polynomial.

In this post, we show that the largest prime factors of the values of this polynomial are “relatively large” infinitely often.

Dirichlet’s Theorem and Sieves

Dirichlet’s Theorem on the infinitude of primes in arithmetic progressions relies on the non-vanishing of non-trivial Dirichlet characters at 1.

In this post, I’ll show how this reduction can be introduced in an intuitive way via sieve theory. If we actually sieve, we obtain estimates for the number of integers whose prime factors lie in given congruence classes.

Sums of Small Divisors

Which integers are a multiple of the sum of their “small” divisors? In the post, we study whether this set should be finite or not.

Our partial solution relates this problem to many outstanding conjectures in number theory about the distribution of prime numbers, such as the twin prime conjecture and the infinitude of Mersenne primes.