I became interested in making music back in the good old days of the Commodore 64 home computer. I listened to the wonderful music of Rob Hubbard, David Whittaker, Ben Daglish, Martin Galway, Fred Gray and all the other musicians who composed for the C64 (I still sometimes find myself whistling the tunes from Spellbound, Master of Magic, Street Surfer, Krakout, Parallax, Shadowfire and other games that I haven't played for at least 20 years).
One of the important – indeed central – aspects of mathematics is the idea of abstracting out interesting structures in order to study those structures in general. The study of groups, rings, fields, vector spaces, category theory, and so on, are examples of this. Other types of abstractions are general methods and principles, such as proof by induction or the Pigeonhole Principle. One such principle caught my attention recently, namely the Inclusion-Exclusion Principle.
Project Euler is a math/programming puzzle web site. I can't remember where I read about it first — perhaps it was in an article on Slashdot? (Edit: Actually, yes, I think it was in this article — which also nails down the date I joined Project Euler: The 3rd of October 2012.) Anyway, the idea is this: Every weekend (more or less) a new problem is published. The problem is usually mathematical in nature, but almost always requires some programming to solve. When you have found the correct answer to the problem (most of the time in the form of a number), you submit it to the site and gain access to a forum where you can discuss the problem and the solution.
The first two mathematical conjectures that I describe in this post go back to the days when I was studying math at the University of Copenhagen. At that time I always had several "mathematical pet projects" that I was thinking about whenever I could find some free time. Usually my projects were generalizations of math problems from my classes, and most of them involved number theory, group theory or combinatorics.