#amicable numbers

My last finish of 2021 has arrived at its destination!

These patterns are the Amicable Number Bookmarks by Suzza Silver of Beautyofmathematics.com, and I’ve wanted to stitch a pair for so long-- I think the idea behind them is brilliant! 

220 and 284 are a pair of numbers with a special relationship: if you add up all the proper divisors of 220, you'll get 284, and if you add up the proper divisors of 284, they make 220-- perfectly symmetrical! A pair of numbers with this property are called 'Amicable Numbers', and so they make for a perfect friendship gift. :) I made one bookmark for my friend @i-ddpej, and kept the other for myself.

We can’t force our inner worlds to collide when even touching each other’s seems impossible. Is it the lack of resonance, then, or the presence of patterns? Or perhaps both—both may explain this habitual, emotional, subjective loneliness that makes me feel like a stranger among others. But no, I am not seeking collision, but affinity—for that rare balance where two inner worlds quietly recognize one another, like numbers that exist apart yet reflect each other completely. So would it be too greedy—too wretched of me—to reach for something… for someone? Yet 220 and 284 exist, proof that harmony can be found, even in rarity

This is the first of hopefully many posts on things which are not on the curriculum but people may find interesting/build their understanding of another topic.

To understand amicable numbers you’ll need a good understanding of factors.

Amicable numbers are two numbers of which the factors (excluding the number itself) of each adds up to the other number in the pair. For example, the lowest of these are 220 and 284, since:

Sum of factors of 220:

1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284

Sum of factors of 284:

1 + 2 + 4 + 71 + 142 = 220!

Ashley and I found this awesome enough that we have keyrings of them from the brilliant website Mathsgear:

A similar concept to amicable numbers is perfect numbers. A perfect number is a number such that all its factors add up to itself. The smallest known example of this is 6:

Sum of factors of 6:

1 + 2 + 3 = 6

And another small example is 28:

Sum of factors of 28:

1 + 2 + 4 + 7 + 14 = 28

Interestingly, relatively few perfect numbers are known as they get bigger very quickly. Only 4 of these were known to the Greeks, with the largest known one being 8128. The next largest after this is 33550336, which was discovered in 1456!

It didn't take me long to come up with a nice (but way too slow answer). I predicted this might be a problem and after learning my lesson with some earlier problems, I decided to do this in baby steps with my computer. I tested it on 10, and it seemed good, 100, it could handle, and then I tried 1000 and it cried about it.

Spoilers below

365 photos, day forty-five

220 and 284 are called amicable numbers. Each is the sum of the factors of the other…

Factors of 220: 1+2+4+5+10+11+20+22+44+55+110=284 Factors of 284: 1+2+4+71+142=220

Historically, amicable numbers represented perfect harmony and love. Lovers carry talismans which have the numbers engraved on them. My boyfriend doesn’t wear jewellery so I wrote them on mugs instead.