Edit: I actually looked through https://news.ycombinator.com/submitted?id=imurray yesterday to see if there was anything we could invite a repost of. (I do this sometimes when looking at which users were first to submit an interesting article.) I didn't see anything, but on a second look https://news.ycombinator.com/item?id=7984992 could be interesting, so you'll get an email inviting you to repost that one, if you want to :)

I wrote an answer there that produced these diagrams from fairly few lines of code.

The question there referred to same now-broken link mentioned above so the origin still unknown.

The same principle is why this looks like a circle/ellipse but it isn't:

O

You could try writing out an addition table and a multiplication table and see if you can find patterns and differences (you can).

The Sieve of Eratosthenes is good.

I asked ChatGPT what branch of mathematics high school algebra is and it suggested the field theory of real numbers. I have since been looking at groups and fields with some enjoyment.

I made a video that I thought of as a play on Ulam's Spiral [2][3] a while back. Instead of marking primes I marked points of the (square) spiral that were x mod n 0. It is sort of silly and maybe a bit confusing.

[1] https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes

[2]https://en.wikipedia.org/wiki/Ulam_spiral

https://www.youtube.com/watch?v=fRzv1SXKCY8 (my video play it slowed down maybe)

One of the more general forms of this is the logarithm. The defining equation of the logarithm is that log(a*b) = log(a) + log(b). This turns multiplication into a nice linear addition problem.

A less general but more powerful transformation exists on the integers. You can factorize any integer into it's prime factors, then multiply by adding together the powers on matching primes. This may seem like more effort for a similar result, but the prime factors of an integer tell you a lot about that integers divisibility, so it's worth the effort.

If you're talking about the interplay between multiplication and addition: Get a degree in pure maths, learn lots of number theory, and cry at how inelegant it is. Trust me, I did it in a past life

That's a repeating pattern. In the case of 3 the pattern has length 1, and it goes (1, 1, 1, ...). In the case of 7 the pattern has length 6, and it goes (1, 3, 2, 6, 4, 5), starting at the units column. If that looks a bit pseudorandom, there's a good reason, it's vaguely similar to an old-fashioned PRNG.

(The pattern starts again with 1 in the millions column, because 142857 * 7 = 999999. Hence 2 million divided by 7 has remainder 2, 3 million divided by 7 has remainder 3, and so on.)

So 371 should be understood as 3*2 + 7*3 + 1*1, which is 28.

Or as another example, 6993 => 6*6 + 9*2 + 9*3 + 3*1, which is 84.