Category: Math

Earlier, on August 6, 2021, I had tried to formalize a mathematical concept that would generalize the idea of a “basis,” as seen in the theory of vector spaces or the Fundamental Theorem of Arithmetic. I discuss in this post what I tried and what I learned.

We consider how we can do algebra with colors. In other words, if we take the set of all colors, what algebraic structures can it be endowed with? In this post, we look at the algebra of color mixing: what is the structure of the algebra that is induced by the operation of color mixing? (What properties does color mixing satisfy?)

We can think of modular arithmetic as amending operations on Z to “wrap around and stay in Zn.” Can we generalize this idea to arbitrary groups, or even general algebras?

Let denote the sequence of primes and denote the arithmetic progression with starting term and common difference . Dirichlet’s Theorem on Arithmetic progressions says that if , then contains infinitely many members of . But the Green-Tao Theorem says that for any , there exist such that contains members of . In some sense, these results look like “inverses” of each other from the perspective of sequence containment: at a high level, given two sequences , one result talks about containing members of , and the other talks about containing members of .

Can we make this analogy a bit more precise?