Perfect, Friendly, and Amicable numbers.

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6 and 28 are both perfect. Coincidentally, they’re also very friendly.

Way back when Pythagoras was roaming the lands, he set up a little club called the Pythagoreans which, admittedly, is a little arse-y of him. This club just sat and thought about numbers all day every day. They did figure quite a lot of things out, including finding the first Perfect numbers.

Pythagoras was thinking about the number Six one day and figured out that the sum of its divisors is equal to itself; such that 1 + 2 + 3 = 6. A couple of years later, he found that 28 also carries this trait- such that 1 + 2 + 4 + 7 + 14 = 28. 

He decided to call these numbers perfect numbers- since they had to be the creation of a God. After all, 6 was the first perfect number, and God spent 6 days creating the Earth and the Heavens etc, and took a day of rest. Also, a lunar cycle occurred every 28 days. It was all far too coincidental, and hence he deemed these numbers to be perfect. The next perfect number is 496, which is where the whole idea of divine intervention falls apart, really. 

I think that the most remarkable thing about Pythagoras working on perfect numbers was that he managed to work it out using roman numerals. That is to say that he figured out that I + II + III = VI, and I * II * III = VI, and he saw that as significant. Very cool.

Interestingly, all perfect numbers are even. While nobody has proven that a perfect number cannot be odd, nobody has found an odd perfect number.

 

And now it gets a little bit more interesting. Friendly numbers (definitely not to be confused with amicable numbers) also exist. These are pairs of numbers whose abundance (The sum of the numbers divisors divided by itself) are the same. For example, 30 and 140 are friendly numbers:

The sum of 30s divisors= 1 + 2 + 3 + 5 + 6 + 10 + 15 + 30 = 72

72 / 30 = 12/5

The sum of 140s divisors= 1 + 2 + 4 + 5 + 7 + 10 + 14 + 20 + 28 + 35 + 70 + 140 = 336

336 / 140 = 12/5 

Hence the abundance of these numbers is the same, and as such they are friendly numbers. How poetic.

 

Lastly, there are a family of numbers known as amicable numbers. I detest that friendly and amicable numbers are two completely different things, despite the similarity in their names. In any case, amicable numbers are pairs of numbers who the sum of their proper divisors are equal to one another.

The first pair of amicable numbers are 220 and 284, so the summations get quite messy at this point… However:

Sum of 220’s divisors = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284

Sum of 284’s divisors = 1 + 2 + 4 + 71 + 142 = 220

Their divisors are equal to each other, and hence they are amicable numbers.

 

All of these sets of numbers are really quite pretty, but they are very few and far apart, so kudos to anyone that can be arsed to work them out. 

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This entry was posted by dailysliceofpi.

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