Amicable

They call certain numbers amicable numbers, adopting virtues and social qualities to numbers, such as 284 and 220; for the parts of each have the power to generate the other. ~Iamblichus of Chalcis

In Rites of Love and Math (yes, I finally got to see it), a mathematician is being hunted down for his great discovery- an equation for love. For some, the idea of love being quantifiable might seem appalling in its lack of romance.

What these individuals might not realize, however, is that math already has some delightfully intimate, romantic aspects to it.

An example of this is amicable numbers.

Amicable numbers are pairs of numbers where each number is the sum of the divisors of the other number (discounting the number itself, which is, of course, a divisor yielding 1). The smallest (and most well-known) amicable pair is 220 (1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110) and 284 (1, 2, 4, 71, 142).

Currently, we know of about 12 million amicable pairs, but we have no perfect formula for figuring them out. Over the years, a few usable (but incomplete) formulas have been discovered. The first was by a ninth-century Arabian mathematician named Thābit ibn Qurra (or Kurrah, depending on the source). Good ol’ Euler then expanded upon that formula, discovering dozens of new pairs (though, fun fact: two of the pairs Euler submitted as amicable pairs were actually wrong). And now, we have computers that have discovered millions.

And yet there’s currently no use for amicable pairs. Their interconnectedness has found no application anywhere in the world of mathematics.

It’s their entwined nature, however, that makes these numbers so intriguing. In fact, these pairs have shown up throughout history. 220 and 284 have been known since the days of Pythagoras. In fact, it is said that Pythagoras himself once described a friend as, “one who is the other I, such as are 220 and 284.”

Whether Pythagoras actually said that or not, amicable numbers have been entrenched in mysticism for hundreds of years. Ozanum, an eighteenth-century French mathematician,  gives examples in his Mathematical Recreations of amicable numbers, remarking that 220 is equal to the sum of the aliquot parts of 284.

In his book, Number Theory and Its History, Oystein Ore said, “In Arab mathematical writings the amicable numbers occur repeatedly. They play a role in magic and astrology, in the casting of horoscopes, in sorcery, in the concoction of love potions, and in the making of talismans.”

Talismans of this sort were worn in the middle ages. They were inscribed with 220 and 284 and worn to promote love or guarantee a perfect friendship between the people who wore them (seeing as love and friendship are also deeply entangled, I suppose that only makes sense). As W. Wynn Westcott wrote, “The parts of each [amicable number] are generative of each other, according to the nature of friendship.” The world’s first friendship necklaces.

And, in what I consider to be utterly romantic, an Arab numerologist documented the practice of carving amicable numbers on two pieces of fruit. You would eat one fruit, then give the other to your lover as a mathematical aphrodisiac.

French poet Marceline Desbordes-Valmore once wrote, “Entre deux coeurs qui s’aiment, nul besoin de paroles (Two hearts in love need no words).” Maybe she was right.

Maybe all they need are the right numbers.

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