# Fractal Sets and Arithmetic Progressions

If a set of positive integers contains no arithmetic progressions, how large can it be? In this post, we study this question in the context of harmonic sums.

If a set of positive integers contains no arithmetic progressions, how large can it be? In this post, we study this question in the context of harmonic sums.

This post discusses the problem of computer-assisted decryption of un-parsed substitution ciphers. Sample Mathematica code is linked within.

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.

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.

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?)

Our modern calendar, the Gregorian calendar, introduced several changes to the Julian calendar, including the introduction of a three-step algorithm for determining leap years.

But just how accurate is the Gregorian model? Can centuries of mathematics and astronomy lead us to a more accurate calendar?

In this post, we’ll look at one-player games of limited information (sometimes classified as puzzles, not games) through a topological lens, and create for each game a poset of topologies under which topologically indistinguishable points correspond to outcomes that are indiscernible in a limited-information context. Expanding this dictionary, we’ll describe a topology on the outcome space under which the “safe” or “warranted” extension of one’s limited information relates to the continuity of certain maps.