Amicable numbers

Published Juni 18, 2026 · By NumberWiki

Two numbers are amicable when each one is exactly the sum of the other's proper divisors. The smallest pair — 220 and 284 — has been known since antiquity, and the search for more of them runs from the Pythagoreans through Fermat, Descartes, and Euler to a teenager who spotted a pair every one of them had missed.

The definition

Recall that the proper divisors of a number are all its positive divisors except itself. Add them up and you get the number's aliquot sum. A perfect number is its own aliquot sum. An amicable pair is the next step out: two different numbers, each equal to the other's aliquot sum.

Take 220 and 284:

The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110 — and they sum to 284.
The proper divisors of 284 are 1, 2, 4, 71, 142 — and they sum to 220.

Each number points at the other. Where a perfect number is "friends with itself," amicable numbers are friends with a partner — which is exactly how the ancients read them.

A symbol of friendship

The pair (220, 284) was known to the Pythagoreans, who invested it with symbolic weight: it stood for friendship, harmony, and mutual goodwill. The association persisted for two thousand years. Medieval Arab and European writers prescribed talismans inscribed with 220 and 284 to secure love; one report has people consuming food marked with the two numbers — one partner eating 220, the other 284 — as a charm for romance. In Genesis, commentators noted, Jacob gives Esau 220 goats (200 she-goats and 20 he-goats) as a gift to win his brother's favor — a number, the medieval scholar Abraham Azulai suggested, chosen for its amicable significance.

Thābit ibn Qurra's formula

For centuries 220 and 284 stood alone. The first real progress came in 9th-century Baghdad, where the polymath Thābit ibn Qurra proved a remarkable rule: for certain values of n, if the three numbers p = 3·2ⁿ⁻¹ − 1, q = 3·2ⁿ − 1, and r = 9·2²ⁿ⁻¹ − 1 are all prime, then 2ⁿ·p·q and 2ⁿ·r form an amicable pair. Setting n = 2 regenerates 220 and 284; n = 4 gives the pair 17,296 and 18,416.

Thābit's formula was studied further by later Islamic mathematicians — al-Fārisī and, in the 17th century, Muhammad Baqir Yazdi, who found the pair 9,363,584 and 9,437,056 long before European mathematics caught up.

Fermat, Descartes, and Euler's avalanche

Working from Thābit's rule, Fermat announced the pair 17,296 / 18,416 in 1636, and Descartes contributed 9,363,584 / 9,437,056 in 1638 — both, unknowingly, rediscoveries of what the Islamic mathematicians already had. Then Leonhard Euler transformed the subject: with more general methods he published a list of 58 amicable pairs in 1750, more than everyone before him combined. For a while it seemed Euler had found essentially all the small ones.

The pair everyone missed

He hadn't. In 1866 a sixteen-year-old Italian, Nicolò Paganini (no relation to the violinist), found the pair 1,184 and 1,210 — the second-smallest amicable pair, small enough that Fermat, Descartes, and Euler could have found it by hand, yet overlooked by all of them for centuries. It is a perfect cautionary tale about assuming the small cases are all accounted for, and it is the kind of discovery that a patient teenager with no special machinery can still make.

Amicable numbers and the aliquot sequence

Amicable pairs are the simplest case of a deeper idea. Start from any number and repeatedly take the aliquot sum — the sum of proper divisors. This aliquot sequence can do several things: fall to zero (most numbers), get stuck on a perfect number (a fixed point), or loop. A loop of length two is an amicable pair. Longer loops are called sociable numbers — the smallest is a 5-cycle starting at 12,496, and there is a famous 28-cycle beginning at 14,316. Whether some aliquot sequences grow forever without ever closing into a loop or hitting zero is a celebrated open problem; the smallest number whose fate is unknown is 276. NumberWiki shows the start of the aliquot sequence on every composite number's page.

What's known, and what isn't

Modern computer searches have found more than a billion amicable pairs. Yet the basic questions remain stubbornly open:

Are there infinitely many amicable pairs? Believed yes, unproven.
Is there an amicable pair where one number is odd and the other even? None has ever been found.
Is there a pair whose two members are coprime (share no common factor)? If one exists, it must be larger than 10⁶⁷.
Every known pair shares a common factor — and all known pairs have both members the same parity (both even, or both odd).

Amicable numbers on NumberWiki

Members of an amicable pair are detected automatically and tagged amicable; the page also shows the number's aliquot sum and the start of its aliquot sequence, so you can watch the two partners point at each other. Related families: perfect numbers (aliquot fixed points), abundant and deficient numbers. The pairs to start with: 220 & 284, 1,184 & 1,210, 2,620 & 2,924.