# Website activity diary 2018

Not the busiest of years, but I at least found a few little curiosities to amuse myself with, and made a little progress on some serious mathematics.

A heinous hack

Wrote up a heinous hack I'd used to code round a problem; thankfully, it's since been retired.

Prime multiplicities in factorials

A neat little result about how many times each prime shows up as a factor in the product of all the positive naturals up to a given one.

The difference between a sphere and a cylinder

A neat little result about how much is left when you cut out a cylindrical core from a sphere.

Tweaked Life

Made my implementation of Conway's cellular automaton a little easier to play with.

Power, exp and ln

Separated power from polynomial and began generalising it.

Differentiate polynomials

Explored the gradients of chords to show how an algebraically formulated deriviative matches my latest reformulation of how to do differentiation. Explored several other details (e.g. multinomials, particularly homogenous ones) in support of that.

Permutations

Began describing one of the tools that shows up repeatedly all over mathematics.

Natural arithmetic
I began examining the consequences of applying general arithmetic to the natural numbers, starting with a look at unique prime factorisation.
Another look at polynomials
Now contrasting their formal treatment, as lists of coefficients under a contrived arithmetic, with their interpretation as functions from a ringlet to itself.
Began describing differentiation
Introducing it in the context of linear spaces, rather than reals, using simplices to generalise chords and provide gradients. I've had this in mind for many years.
Reworked positive ordering
A clarification of earlier work on when arithmetic itself induces order on its values.
Earlier history
2017, 2016, 2011–2015, 2010, 2009, 2008, 2007, 2006.
Written by Eddy.