Curious Matheology (or the Number of Reconciliation)

I've been reading this book about mathematics by Paul Hoffman called Archimedes Revenge. At the diner table today I started to tell my kids about some of the fascinating kernels of math trivia I've gleaned from it, when they told me they'd rather read about it on my blog. Around here this is generally code for "spare me, Dad," but I'm going to assume they were sincere, and offer them here.

Here's one to warm up with: the ancient mathematician Pythagoras held that any number that was the sum of its own factors (excepting the number itself) was a "perfect" number. The factors of the number 28, for instance (1, 2, 4, 7 and 14) all add up to 28. Any Creationism buffs out there want to guess what's the first perfect number?

And then there's this one: in his account of the post-resurrection miraculous catch of fish (John 21:1-14), John notes very carefully that there were precisely 153 fish in the net. The disciples are about to eat their first meal with the Resurrected Lord of Life, and someone had the time and temerity to stop and count the fishes? For millenia scholars have wondered if there isn't something more going on here.

According to the JewishEncyclopedia.com, the Tetragrammaton (the Hebrew covenant name for God) appears 153 times in the book of Genesis (this is suggestive, though when I did a search on e-sword, I only turned up 142... maybe I'm missing something). But let's add this: in Hebrew Gematria (an esoteric system of determining the "number" of various Hebrew names and phrases), the number of the Hebrew phrase "sons of God" adds up to 153 (check it out here).

This is all curious enough, but consider these mathematical tidbits. For starters, 153 is a triangular number (and it smaller components, 1 and 15, are also triangular numbers). It's also the smallest number that can be expressed as the sum of the cubes of its own digits (which is to say that: 1^3 + 5^3 + 3^3= 1 + 125 + 27= 153.) What's more, they say that 153 "lies dormant" in every third number. (Take any multiple of three, sum the cubes of its digits, take the result and do the same, and do the same, and do the same, and eventually you will reach 153). What does all this have to do with Jesus and a miraculous catch of fish that proved he was the Resurrected Lord? I'm not really sure, except that there are enough 3's in there to make you go hmmm.

But that's all just brain sit-ups for this one. In Genesis 32:14, when Jacob meets Esau after a long, sordid history of sibling rivalry, Jacob gives his brother a set of 220 goats (200 female, 20 male) and 220 sheep (200 ewes, 20 rams), as a gesture of reconciliation.

Paul Hoffman points out that the sum of all the numbers that divide evenly into 220 (1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110) is 284, and the sum of all the divisors of 284 (1, 2, 4, 71, and 142) is 220. In other words, 220 and 284 are each equal to the sum of the divisors of the other.

Now: anyone want to guess what the ancient Pythagoreans called a pair of numbers where each is equal to the sum of the divisors of the other?

Friendly numbers. Go figure: 220 is the smallest known friendly number.

To be sure, Pythagoras was a Greek Mathematician and the author of Genesis was an ancient Hebrew, and who-knows-how-many centuries separated them. But there's still something, well, intruiging at least, about the fact that Jacob chose a "friendly number" in making his peace offering of sheep and goats to his brother.

Of course, the total number of livestock in Jacob's gift to Esau was actually 580, not 220, making Hoffman's whole reading of the story specious. So: anyone know any esoteria about the number 580?