Amicable numbers
Amicable numbers are pairs of distinct positive integers (m, n) such that the sum of the proper divisors of m equals n, and the sum of the proper divisors of n equals m, where proper divisors exclude the number itself.[1] The smallest such pair is (220, 284), where the proper divisors of 220 sum to 284 (1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284), and those of 284 sum to 220 (1 + 2 + 4 + 71 + 142 = 220).[1][2] These numbers have a rich history dating back to ancient times, with the pair (220, 284) reportedly known to Pythagoras around 500 BCE and used in mystical contexts such as astrology and love potions in Pythagorean traditions.[2] In the 9th century, the Arab mathematician Thābit ibn Qurra developed a rule for generating certain amicable pairs: if p = 3 \cdot 2^{n-1} - 1, q = 3 \cdot 2^n - 1, and r = 9 \cdot 2^{2n-1} - 1 are all prime for some integer n > 1, then $2^n \cdot p \cdot q and $2^n \cdot r form an amicable pair.[2] This method yielded pairs like (17,296, 18,416) for n=4, rediscovered by Fermat in 1636, and (9,363,584, 9,437,056) for n=7, noted by Descartes in 1638.[1][2] Euler cataloged 59 valid pairs by 1747 (with five errors later corrected), and the second-smallest pair (1,184, 1,210) was found by 16-year-old Niccolò Paganini in 1866.[1][2] Key properties distinguish amicable numbers from perfect numbers, where the sum of proper divisors equals the number itself; amicable pairs are "mutually perfect" but the numbers are unequal.[1] All known pairs have even sum m + n, and their divisor sums satisfy \sigma(m) = \sigma(n) = m + n, where \sigma is the sum-of-divisors function.[1] Pairs are classified as regular (generated by rules like Thābit's or Euler's) or irregular, with no known pairs coprime to 210 (i.e., not divisible by 2, 3, 5, or 7).[1] It remains unknown whether infinitely many amicable pairs exist, though their density is zero, as proven by Erdős.[3] As of November 2025, over 1.2 billion amicable pairs are known from distributed computing efforts, with the smallest member of the largest known pair exceeding $10^{20}, and ongoing searches extending beyond $10^{21}.[4] No complete enumeration is possible, and open questions persist regarding odd amicable pairs (none known) and pairs relatively prime to each other (none known below approximately $10^{25} if they exist).[2][1]Definition and Properties
Definition
Amicable numbers are two distinct positive integers m and n, with m < n, such that the sum of the proper divisors of m equals n and the sum of the proper divisors of n equals m.[5] Proper divisors of a number are all its positive divisors excluding the number itself.[5] In standard mathematical notation, the sum-of-divisors function \sigma(k) gives the sum of all positive divisors of k, so the proper divisor sum is s(k) = \sigma(k) - k; thus, m and n form an amicable pair if s(m) = n, s(n) = m, and m \neq n.[5] The smallest known amicable pair is (220, 284).[6] The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, which sum to 284.[7] The proper divisors of 284 are 1, 2, 4, 71, and 142, which sum to 220.[7] Unlike perfect numbers, where s(k) = k, amicable numbers involve a mutual exchange between two distinct numbers, with each being either abundant (s(k) > k) or deficient (s(k) < k) relative to itself but balancing the pair.[5] In antiquity, such pairs were believed to possess mystical properties and were associated with Pythagorean traditions.[8]Fundamental Properties
Amicable pairs (m, n) with m < n satisfy \sigma(m) = \sigma(n) = m + n, where \sigma(k) denotes the sum of all positive divisors of k.[9] This equality follows directly from the definition using the proper divisor sum function s(k) = \sigma(k) - k, since s(m) = n implies \sigma(m) = m + n and s(n) = m implies \sigma(n) = n + m. Adding these relations yields \sigma(m) + \sigma(n) = 2(m + n). To see this, start with s(m) = n and s(n) = m, so \sigma(m) - m = n and \sigma(n) - n = m. Adding the equations gives \sigma(m) - m + \sigma(n) - n = n + m, which rearranges to \sigma(m) + \sigma(n) = 2(m + n).[9] Neither member of an amicable pair can be prime. If p is prime, then s(p) = 1, but s(1) = 0 \neq p, so no prime can pair with another number to form an amicable pair.[6] In an amicable pair (m, n) with m < n, m is abundant since s(m) = n > m, while n is deficient since s(n) = m < n. For the smallest pair (220, 284), s(220) = 284 > 220 confirms 220's abundance, and s(284) = 220 < 284 confirms 284's deficiency.[1] Amicable pairs exhibit consistent parity: all known pairs have both members even or both odd, with the first odd pair discovered by Leonhard Euler in the 18th century. No mixed-parity pairs are known, and it remains open whether they exist.[1][10]Historical Development
Ancient and Early Discoveries
The concept of amicable numbers, pairs of distinct positive integers where each equals the sum of the proper divisors of the other, traces its origins to the Pythagorean school around 500 BCE. The Pythagoreans, who viewed numbers as embodying mystical and cosmic principles, are credited with early recognition of such pairs for their symbolic representation of harmony and reciprocity, though no specific examples survive from their writings.[11] This attribution stems from later accounts, highlighting the Pythagoreans' broader fascination with numerical relationships as manifestations of divine order.[12] The earliest documented pair, 220 and 284, was recorded by the Neoplatonist philosopher Iamblichus (c. 245–325 CE) in his commentary on Nicomachus of Gerasa's Introduction to Arithmetic. Iamblichus, drawing on Pythagorean traditions, described these numbers as "friendly" due to their mutual affection: the proper divisors of 220 (1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110) sum to 284, while those of 284 (1, 2, 4, 71, and 142) sum to 220.[13] This presentation framed amicable numbers within a philosophical context of numerical friendship, preserving and elaborating on earlier oral or lost Pythagorean lore.[14] In the broader ancient Greek intellectual landscape, amicable numbers aligned with numerological explorations of harmony, as seen in Theon of Smyrna's Mathematics Useful for Reading Plato (c. 100 CE), which delved into Pythagorean classifications of numbers and their proportional relationships to illustrate cosmic balance.[15] No verified amicable pairs predate Iamblichus' account.Medieval and Early Modern Contributions
During the Islamic Golden Age, the mathematician Thābit ibn Qurra (c. 836–901 CE) made significant advancements in the study of amicable numbers by developing a systematic rule for generating such pairs, building on earlier Greek interest. He is credited with discovering the amicable pair (17,296, 18,416), which was the first new pair identified since antiquity, demonstrating a deeper understanding of divisor sums and their properties.[7][16] This contribution, preserved in Arabic mathematical texts, highlighted the potential for constructing amicable numbers through specific prime factorizations, though the full details of his method were rediscovered centuries later. In the early modern period, European mathematicians independently revived and extended this work amid renewed interest in number theory. Pierre de Fermat (1607–1665) rediscovered Thābit's rule around 1636 and applied it to find the same pair (17,296, 18,416), communicating his results through correspondence facilitated by Marin Mersenne, which spurred further exploration. Similarly, René Descartes (1596–1650) in 1638 used an analogous approach to identify the much larger pair (9,363,584, 9,437,056), showcasing the rule's capacity to produce increasingly complex examples without prior knowledge of Thābit's original formulation.[16][17] These discoveries, exchanged among scholars like Mersenne, marked a transition from isolated findings to collaborative inquiry in Europe. Leonhard Euler (1707–1783) brought a more rigorous and comprehensive approach in the 18th century, systematizing the study of amicable numbers through the divisor sum function (σ(n)). In works such as E100 (1747) and E152 (1750), he cataloged 59 new pairs, including examples like (2,620, 2,924), by systematically computing σ values and identifying reciprocal relationships. Euler also generalized Thābit's rule, expanding its scope and establishing foundational techniques that influenced subsequent generations, though he built directly on the pairs from Fermat and Descartes.[16][17]Methods of Generation
Thābit ibn Qurrah's Theorem
Thābit ibn Qurrah formulated a criterion in the 9th century for generating certain amicable pairs using a specific construction involving powers of 2 and presumed primes.[18] The theorem states that for an integer n > 1, if the numbers p = 3 \times 2^{n-1} - 1, q = 3 \times 2^n - 1, and r = 9 \times 2^{2n-1} - 1 are all prime, then m = 2^n \cdot p \cdot q and k = 2^n \cdot r form an amicable pair, meaning the sum of the proper divisors of m equals k and vice versa.[18][19] This construction yields known amicable pairs only for n = 2, 4, 7. For n=2, p=5, q=11, r=71 (all prime), giving the pair (220, 284). For n=4, p=23, q=47, r=1151 (all prime), producing (17296, 18416). For n=7, p=127, q=383, r=9349 (all prime), resulting in the pair (9363584, 9437056).[18][19] No other values of n up to very large limits satisfy the primality conditions for p, q, and r simultaneously, though extensive computational searches have confirmed the absence of additional pairs from this rule for very large n.[18] The proof relies on the multiplicativity of the divisor sum function \sigma, which for coprime factors multiplies accordingly. Consider m = 2^n \cdot p \cdot q, where p and q are distinct odd primes. Then, \sigma(m) = \sigma(2^n) \cdot \sigma(p) \cdot \sigma(q) = (2^{n+1} - 1) \cdot (p + 1) \cdot (q + 1). Substituting the forms, p + 1 = 3 \times 2^{n-1} and q + 1 = 3 \times 2^n, yields \sigma(m) = (2^{n+1} - 1) \cdot (3 \times 2^{n-1}) \cdot (3 \times 2^n) = (2^{n+1} - 1) \cdot 9 \times 2^{2n-1}. For k = 2^n \cdot r with r prime, \sigma(k) = (2^{n+1} - 1) \cdot (r + 1). Since r + 1 = 9 \times 2^{2n-1}, it follows that \sigma(k) = (2^{n+1} - 1) \cdot 9 \times 2^{2n-1} = \sigma(m). To confirm amicability, verify \sigma(m) = m + k, which holds because m + k = 2^n (p q + r), and algebraic expansion using the definitions of p, q, and r equates this to (2^{n+1} - 1) \cdot 9 \times 2^{2n-1}. Thus, the sum of proper divisors of each is the other number.[18][19] The theorem's limitations stem from the rarity of simultaneous primality for p, q, and r as n increases; for example, at n=3, r=287=7 \times 41 is composite, and at n=5, q=95=5 \times 19 is composite. This scarcity means the rule generates only three known pairs, inspiring later generalizations like Euler's but producing few instances itself.[18]Euler's Rule and Extensions
Leonhard Euler significantly advanced the constructive generation of amicable pairs in the 18th century by generalizing earlier methods and systematically applying properties of the divisor sum function \sigma. His approach focused on pairs of the form $2^n \cdot p \cdot q and $2^n \cdot r, where p, q, and r are distinct odd primes, and n \geq 1. For such numbers to be amicable, the primes must satisfy \sigma(2^n \cdot p \cdot q) = 2^n \cdot r and \sigma(2^n \cdot r) = 2^n \cdot p \cdot q, which simplifies using the multiplicativity of \sigma: specifically, \sigma(2^n \cdot k) = (2^{n+1} - 1) \sigma(k) for odd k. This leads to the conditions r = p q + p + q (all prime) and $2^n = (p q + p + q + 1)/(p + q + 2).[20][17] Euler further parameterized this rule using integers n > m > 0, defining: p = 2^m (2^{n-m} + 1) - 1, q = 2^n (2^{n-m} + 1) - 1, r = 2^{n+m} (2^{n-m} + 1)^2 - 1. If p, q, and r are all prime, then $2^n p q and $2^n r form an amicable pair. This formulation encompasses previous results as a special case when m = n-1, recovering Thābit ibn Qurrah's theorem, but allows broader exploration by varying m and n. A representative example occurs for n=2, m=1: p=5, q=11, r=71 (all prime), yielding the pair (2^2 \cdot 5 \cdot 11, 2^2 \cdot 71) = (220, 284). Using this and related techniques, Euler discovered 59 new amicable pairs, expanding the known list from three to over 60 by 1750.[21][20] Extensions of Euler's rule accommodate greater structural variety in the odd parts of the numbers, such as products of more than two primes or higher powers of primes, by solving analogous equations from \sigma(2^a u) = 2^b v and \sigma(2^b v) = 2^a u for odd u, v and possibly unequal exponents a \neq b. Euler himself outlined paths to such generalizations, noting that amicable pairs could arise from forms like a \cdot p \cdot q and a \cdot r \cdot s (with a a common factor and additional primality conditions on r+1 = (p+1)(q+1) or similar), or by incorporating prime powers beyond the first degree in the factorization. These refinements enabled the discovery of dozens more pairs in the 18th and 19th centuries, including cases with squared factors like the pair (1184, 1210) = (2^5 \cdot 37, 2 \cdot 5 \cdot 11^2), found in 1866 despite fitting an extended framework akin to Euler's methods. Modern variants continue this by systematically testing higher powers and additional prime factors computationally, producing many known pairs while emphasizing the rule's role in establishing inverse relations between the odd parts' divisor sums.[22][17][21]Specific Classes of Pairs
Smallest Known Pairs
The smallest known amicable pair, ordered by the smaller number, consists of 220 and 284. The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, which sum to 284. The proper divisors of 284 are 1, 2, 4, 71, and 142, which sum to 220.[1] The next smallest pair is 1184 and 1210. The proper divisors of 1184 are 1, 2, 4, 8, 16, 32, 37, 74, 148, 296, and 592, summing to 1210. The proper divisors of 1210 are 1, 2, 5, 10, 11, 22, 55, 110, 121, 242, and 605, summing to 1184.[23][24][25] The third smallest pair is 2620 and 2924. The sum of the proper divisors of 2620 is 2924, and the sum of the proper divisors of 2924 is 2620.[23][26][27] The fourth and fifth smallest pairs are 5020 and 5564, and 6232 and 6368, respectively, each satisfying the amicable condition where the proper divisor sum of the first equals the second and vice versa.[23] Among the earliest known amicable pairs, the pattern is mixed: the first six pairs consist of two even numbers each, the seventh pair consists of two odd numbers, and the eighth pair consists of two even numbers.[23] For completeness, the first ten smallest amicable pairs, ordered by the smaller member, are listed below:| Index | Smaller Number | Larger Number |
|---|---|---|
| 1 | 220 | 284 |
| 2 | 1184 | 1210 |
| 3 | 2620 | 2924 |
| 4 | 5020 | 5564 |
| 5 | 6232 | 6368 |
| 6 | 10744 | 10856 |
| 7 | 12285 | 14595 |
| 8 | 17296 | 18416 |
| 9 | 63020 | 76084 |
| 10 | 66928 | 66992 |
Regular and Twin Amicable Pairs
Regular amicable pairs form a subclass of amicable pairs characterized by a specific structural form that facilitates their generation through algebraic rules. An amicable pair (m, n) with m < n is regular if, letting g = \gcd(m, n), both m/g and n/g are square-free positive integers coprime to g. This condition ensures that the odd parts of m and n (after factoring out the common power of 2 in g) have no squared prime factors and share no primes with g, often resulting in pairs composed of a power of 2 multiplied by a small number of distinct odd primes.[28] Such pairs are typically easier to construct using methods like Thābit ibn Qurrah's theorem or its extensions, as the square-free requirement aligns with the multiplicative properties exploited in these rules.[29] The smallest regular amicable pair is (220, 284), where g = 4 = 2^2, $220/4 = 55 = 5 \times 11 (square-free and coprime to 4), and $284/4 = 71 (prime, hence square-free and coprime to 4); this pair corresponds to type (2,1), indicating two odd primes in the smaller member's odd part and one in the larger.[28] Another early example is (17296, 18416), attributed to Thābit ibn Qurrah, with g = 16 = 2^4, $17296/16 = 1081 = 23 \times 47, and $18416/16 = 1151 (prime); also type (2,1).[28] Regular pairs often exhibit one abundant member (the smaller) and one deficient member (the larger), mirroring general amicable properties but with constrained prime factorizations that limit their abundance.[29] Further examples of regular amicable pairs include those discovered by later mathematicians using similar constructive methods. For instance, (9363584, 9437056) is a type (2,1) pair identified by Descartes (though initially containing a composite factor mistaken for prime), with g = 2^{12}, the odd part of the smaller being $17 \times 257 (both Fermat primes), and the larger's odd part a single prime.[28] Other known regular pairs, such as those from Euler's parametrizations, yield more examples; extensions of these rules generate pairs of higher types, including type (4,3). The largest known regular pair has 5577 digits and was found by García in 1997 via Wiethaus's rule.[28][29] These examples highlight how regular pairs prioritize minimal prime sets, often involving Fermat primes or Mersenne-like forms, making them prominent in historical developments despite comprising only a subset of all known pairs.| Pair | Type | Discoverer/Notes | Source |
|---|---|---|---|
| (220, 284) | (2,1) | Ancient/Pythagoreans | [28] |
| (17296, 18416) | (2,1) | Thābit ibn Qurrah (c. 9th century) | [28] |
| (9363584, 9437056) | (2,1) | Descartes (17th century) | [28] |
| Examples from Euler extensions | (4,3) | Euler extensions | [29] |
| Large pair (5577 digits) | Varies | García (1997), via Wiethaus rule | [28] |
| Pair | Difference | Notes | Source |
|---|---|---|---|
| (220, 284) | 64 | Smallest overall, also regular | [30] |
| (1184, 1210) | 26 | Second pair | [30] |
| (2620, 2924) | 304 | Third pair | [30] |
| (5020, 5564) | 544 | Fourth pair | [30] |
| (6232, 6368) | 136 | Fifth pair | [30] |
| (10744, 10856) | 112 | Sixth pair | [30] |
| (12285, 14595) | 2310 | Seventh pair, smallest odd | [30] |
| (64112960650, 64128831350) | 15870700 | Largest among first 3,000 pairs | [30] |