Amicable Numbers Calculator (Pairs, Sociable Chains)

Amicable numbers calculator: pair check, partner search, list up to N, sociable chains of iterated aliquot cycles and historical Thabit ibn Qurra formula.

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Amicable numbers

Two positive integers m and n are amicable if the sum of proper divisors of one equals the other, and vice versa: s(m) = n and s(n) = m. The most famous pair is (220, 284): s(220) = 1+2+4+5+10+11+20+22+44+55+110 = 284 and s(284) = 1+2+4+71+142 = 220. Pythagoreans (6th century BC) considered them symbols of perfect friendship; two friends each carried 220 and 284 as talismans.

Formal definition

Let sigma(n) = sum of all divisors of n, s(n) = sigma(n) - n = proper divisor sum (aliquot). Then (m, n) with m != n are amicable iff:

sigma(m) = sigma(n) = m + n

Equivalently: s(m) = n and s(n) = m. If m = n, condition becomes s(m) = m, i.e., m is perfect - degenerate self-amicability case.

Thabit ibn Qurra formula (850 AD)

Arab mathematician Thabit ibn Qurra (826-901) discovered the first explicit formula:

If p = 3 * 2^k - 1, q = 3 * 2^(k-1) - 1, r = 9 * 2^(2k-1) - 1 all prime, then (2^k * p * q, 2^k * r) is an amicable pair.

k=2: p=11, q=5, r=71 all prime -> (220, 284).

k=4: p=47, q=23, r=1151 prime -> (17296, 18416), attributed to Fermat 1636.

k=7: p=383, q=191, r=73727 prime -> (9363584, 9437056), Descartes 1638.

For k in {1, 3, 5, 6, 8, ...}: one of three not prime, Thabit inapplicable.

Discovery history

  • 220, 284: Pythagoreans, ~500 BC.
  • 17296, 18416: Thabit ibn Qurra ~860 AD (formally), Fermat 1636 (rediscovery).
  • 9363584, 9437056: Descartes 1638.
  • 1184, 1210: shocking discovery by 16-year-old Nicolo Paganini in 1866. Missed by Euler who found over 60 pairs.
  • Euler (1747): 30 new pairs, bringing total to 36.
  • Modern computing (GIMPS-style): over 1 billion pairs known as of 2026.

Sociable chains

Generalization of amicability to cycles of length > 2: a sociable chain is a sequence n_1 -> n_2 -> ... -> n_k -> n_1 where each n_{i+1} = s(n_i). If k=1, perfect number. If k=2, amicable pair. If k >= 3, proper sociable chain.

  • Length 5 chain: (12496, 14288, 15472, 14536, 14264) discovered by Poulet 1918.
  • Length 28 chain: discovered 1965 (Poulet), starts with 14316.
  • Chains of length 4, 6, 8, 9, 10 exist but no length 3 found.
  • Conjecture: no length-3 chains exist (open).

Five calculator modes

  1. Pair check: given m, n, checks if amicable.
  2. Partner search: given m, finds amicable partner (or identifies if perfect).
  3. Pair list: all amicable pairs up to N.
  4. Sociable chain: iterated s orbit with cycle detection.
  5. Thabit formula: for given k, checks if p, q, r prime and builds pair.

Notable properties

  • If (m, n) amicable, one is abundant and other is deficient.
  • All known amicable pairs are both even or both odd. Mixed parity: open.
  • Sum of amicable pair always divisible by 9 for pairs with m+n < 10^7 (Lee-Madachy 1972 conjecture).
  • Smallest amicable pair for each prime factor p.
  • Asymptotic density of amicable pairs unknown; conjectured o(n / log n).

Applications

  • Number theory: connections to sigma, Mersenne, arithmetic formulas.
  • History of mathematics: Pythagorean friendship symbol, Arab tradition, European Renaissance.
  • Distributed computing: amicable pair search projects.
  • Cryptography: some pairs appear as examples in factorization exercises.
  • Philosophy: recurring theme in numerology and mystical mathematics.
  • Education: accessible exercise introducing divisors, sigma, sequences.

Amicable numbers in culture and mysticism

The pair (220, 284) has fascinated humanity for millennia, far beyond pure mathematics. The Pythagoreans saw in amicable numbers a symbol of friendship and harmony, attributing an almost sacred value to the concept: it is said that Pythagoras, asked what a friend was, replied "one who is the other I, like 220 and 284". In the Middle Ages these numbers appeared on talismans and love charms: it was believed that writing 220 and 284 on two objects given to two people would strengthen their bond of affection. Jewish tradition links the pair to the biblical account in which Jacob offers Esau 220 goats as a gift of reconciliation. This rich symbolism shows how an arithmetical curiosity can cross philosophy, religion and superstition.

Open questions about amicable numbers

Despite centuries of study, amicable numbers retain unsolved mysteries. It is not known whether infinitely many amicable pairs exist: billions are now known thanks to computers, but a proof that they are infinite is still lacking. No amicable pair has ever been found in which the two numbers are coprime (with no common divisors), and it is suspected that none exists. All known pairs, moreover, have the same parity: both even or both odd, never a mixed one. Many conjectures about these patterns remain open, and Thabit's formula, though elegant, generates only a tiny fraction of all existing pairs. Amicable numbers remain fertile ground for number theory.

Updated July 2026

Frequently Asked Questions

What are amicable numbers?

Two numbers are amicable if each equals the sum of the proper divisors of the other. The most famous pair is (220, 284): the proper divisors of 220 sum to 284, and those of 284 sum to 220. They are a generalisation of perfect numbers, where a number is amicable with itself. Rare and fascinating, they have been studied since antiquity and carry a strong symbolic value of friendship.

What is the smallest amicable pair?

It is the pair (220, 284), known since the time of the Pythagoreans. The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, which sum to 284; the proper divisors of 284 are 1, 2, 4, 71, 142, which sum to 220. The next pairs are (1184, 1210), (2620, 2924) and (5020, 5564). They rapidly become rarer and larger as the values grow.

What is Thabit ibn Qurra's formula?

It is a rule discovered by the Arab mathematician Thabit ibn Qurra in the 9th century to generate amicable pairs. It states that if the three numbers p = 3·2ᵏ−1, q = 3·2ᵏ⁻¹−1 and r = 9·2²ᵏ⁻¹−1 are all prime, then (2ᵏ·p·q, 2ᵏ·r) is an amicable pair. For k = 2 it gives precisely the pair (220, 284). It is elegant but generates only a small part of all existing pairs.

What are sociable chains?

They are a generalisation of amicable numbers to longer cycles. In a sociable chain, the sum of the proper divisors of each number gives the next, and at the end you return to the first, closing the cycle. Perfect numbers are cycles of length 1, amicable pairs of length 2. There are chains of length 4, 5, 6, 9 and even 28: the most famous starts from 14316 and has 28 elements.

Who discovered amicable numbers?

The pair (220, 284) was already known to the Pythagoreans in the 6th century BC. The Arab mathematician Thabit ibn Qurra formulated a generating rule in the 9th century. In Europe they were rediscovered in the 17th century by Fermat and Descartes, who found new pairs. But the great leap is due to Leonhard Euler, who in the 18th century found as many as 59, revolutionising knowledge of these numbers before the computer age.

Are there infinitely many amicable pairs?

It is not known: it is an open question in number theory. Thanks to computers billions are now known, and ever larger pairs keep being found, but no one has yet proved they are infinite. Many mathematicians think it likely, but a rigorous proof is lacking. It is one of the many apparently simple problems about integers that have resisted a solution for centuries.

What was Paganini's discovery?

In 1866 a young sixteen-year-old Italian named Nicolò Paganini (namesake of the famous violinist) discovered the amicable pair (1184, 1210), which had eluded mathematicians like Fermat and Euler for centuries despite being relatively small. It is the second smallest pair of all, and its late discovery by an amateur is a famous anecdote in the history of number theory.

Can amicable pairs be coprime?

None is known: all known amicable pairs share at least one common factor. It is conjectured that no coprime amicable pairs exist, that is with the two numbers having no common divisors, but it has not been proved. Similarly, all known pairs have the same parity, both even or both odd. These as yet unexplained patterns make amicable numbers an active research topic.

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