I just took on a problem that I found to be simple to understand yet for some reason is still unsolved. Simple to understand turned out to be crucial because when I started I had no idea how much "I did not know" and for that reason I underestimated the problem which is a GOOD thing. It allowed me to slowly grow with the problem over time and learn more about what I need to do to get to where I want to.
The most important part for me is to find a problem that is interesting enough that will keep sucking you back in. Prime factorization is a great example because it is a simple problem so you will continuously make progress and actually feel/see your personal math improvement from week to week.
Initially, I started off by writing down all the numbers up to N to see if I could find some pattern. 20 mins in after some HTML and PHP I had this: https://imgur.com/a/7EhdI (primes in blue, everything else in white)
If you look closely you can see the diagonal shape of the primes. If you try changing the number of columns from 5 to any other number, you will get a distinct shape. The shapes sucked me in, why do they occur? wtf is up with them?! this is so cool yet random.
After having my curiosity tickled, I opened up an excel sheet and started experimenting by throwing these numbers/positions/order/whateverICanThinkOf together in an attempt to find some pattern. Currently, I have an excel file that consists of 35 sheets with different experiments on each sheet. Most experiments and ideas will fail, some will give you a result. But one thing is for sure, I have learned something from each one of them, especially when it comes to algebra. I just get an idea and try it. If I need some help I ask around or google for some info.
The latest idea/experiment that I did showed me that composite numbers do not occur in a random position along the axis of numbers. In fact, numbers come in chains that are related to each other "geometrically".
That's it really. It's like throwing whatever you can think of at a problem to see what sticks. While attempting to do that I slowly learned algebra, a bit of genometry, a bit of statistics and then some. Been fun really and that's all that counts :D
