Amicable Numbers Finder

Discover amicable number pairs and explore the fascinating world of number theory. Check if two numbers are amicable or find all pairs within a range.

Definition: Two numbers (a, b) are amicable if σ(a) - a = b and σ(b) - b = a, where σ(n) is the sum of all divisors of n.

Equivalently: sum of proper divisors of a equals b, and sum of proper divisors of b equals a.

Enter the first number to check
Enter the second number to check
220 & 284
1184 & 1210
2620 & 2924
5020 & 5564
6232 & 6368
6 (Perfect)
30 & 40 (Not Amicable)

Note: This may take longer for larger ranges. The algorithm finds all amicable pairs where both numbers are within the specified range.

Lower bound of the search range
Upper bound of the search range
Limits the number of pairs returned for performance
1 to 1000
1 to 10000
10000 to 50000
1 to 100000

Explore: View known amicable pairs and their properties. Click on any pair to analyze it further.

Thābit ibn Qurra's Theorem (9th century):

For n > 1, let:

p = 3 × 2n-1 - 1

q = 3 × 2n - 1

r = 9 × 22n-1 - 1

If p, q, and r are prime numbers, then 2n × p × q and 2n × r are amicable numbers.

Source: Thābit ibn Qurra, "Book on the Determination of Amicable Numbers" (9th century)
Enter n > 1 (try 2, 4, 5, 6, 7, 8, 9, 10)
Maximum value to check for prime numbers
n=2 (220, 284)
n=4 (17296, 18416)
n=5 (9363584, 9437056)
n=7 (Test)
Calculating...

Understanding Amicable Numbers

Amicable numbers are two different numbers related in such a way that the sum of the proper divisors of each is equal to the other number. They have fascinated mathematicians since ancient times.

Mathematical Definition:

Two numbers (a, b) are amicable if:

σ(a) - a = b and σ(b) - b = a

where σ(n) is the sum of all positive divisors of n, and σ(n) - n is the sum of proper divisors (excluding n itself).

Historical Background

1

Ancient Origins: The smallest pair (220, 284) was known to the ancient Greeks. Pythagoras considered them to symbolize friendship and harmony.

Source: Iamblichus, "Life of Pythagoras" (c. 300 CE)
2

Islamic Mathematics: Arab mathematicians made significant contributions. Thābit ibn Qurra (826-901) formulated a theorem for generating amicable numbers.

Source: Thābit ibn Qurra, "Book on the Determination of Amicable Numbers" (9th century)
3

Modern Discoveries: With computers, millions of amicable pairs have been discovered. The largest known pairs have over 240,000 digits each.

Source: Chermoni & Garcia, "New Amicable Pairs of Record Size" (2021)

Thābit ibn Qurra's Theorem

Theorem: For n > 1, let p = 3 × 2ⁿ⁻¹ - 1, q = 3 × 2ⁿ - 1, and r = 9 × 2²ⁿ⁻¹ - 1. If p, q, and r are prime numbers, then 2ⁿ × p × q and 2ⁿ × r are amicable numbers.

For n = 2: p = 5, q = 11, r = 71 (all prime) → 2² × 5 × 11 = 220 and 2² × 71 = 284.

Source: Thābit ibn Qurra, "Book on the Determination of Amicable Numbers" (9th century)

Types of Number Relationships

Type Definition Example References
Perfect Numbers σ(n) - n = n (equals its own aliquot sum) 6, 28, 496 Euclid, "Elements" (c. 300 BCE)
Amicable Numbers σ(a) - a = b and σ(b) - b = a 220 & 284 Iamblichus, "Life of Pythagoras" (c. 300 CE)
Sociable Numbers Chain of 3 or more numbers where each equals the sum of divisors of the previous 12496 → 14288 → 15472 → 14536 → 14264 → 12496 Poulet, "Les nombres sociables" (1918)
Betrothed Numbers σ(a) - a - 1 = b and σ(b) - b - 1 = a 48 & 75, 140 & 195 Cohen & te Riele, "On Betrothed Numbers" (1995)

Sociable Numbers

Definition: A sequence of numbers where each number is the sum of the proper divisors of the previous number, and the sequence returns to the starting number after k steps (where k ≥ 3).

Example (4-cycle):

1264460
1547860
1727636
1305184
1264460

This is a sociable 4-cycle discovered by Poulet in 1918.

Source: Poulet, "Les nombres sociables" (1918)

Betrothed Numbers

Definition: Two numbers (m, n) such that the sum of the proper divisors of m is one more than n, and the sum of the proper divisors of n is one more than m.

Formally: σ(m) - m - 1 = n and σ(n) - n - 1 = m

Examples: (48, 75), (140, 195), (1050, 1925)

Also called "quasi-amicable numbers" or "reduced amicable numbers".

Source: Cohen & te Riele, "On Betrothed Numbers" (1995)

Applications of Amicable Numbers

  • Cryptography: Some encryption methods use amicable numbers in key generation algorithms
  • Number Theory: Study of divisor functions and their properties
  • History of Mathematics: Illustrates the development of mathematical thought across cultures
  • Recreational Mathematics: Puzzles and mathematical curiosities
  • Education: Teaching concepts of divisors, sums, and number relationships

Calculator Features:

  • Check if any two numbers are amicable
  • Find all amicable pairs within a specified range
  • Explore known historical amicable pairs with references
  • Generate amicable pairs using Thābit ibn Qurra's theorem
  • Check for perfect, sociable, and betrothed numbers
  • Display proper divisors and their sums
  • Export results as CSV or JSON
  • Calculate with efficient algorithms for large ranges

Frequently Asked Questions

A perfect number equals the sum of its own proper divisors (e.g., 6 = 1+2+3). Amicable numbers come in pairs where each number equals the sum of the proper divisors of the other (e.g., 220 and 284). Perfect numbers can be considered "self-amicable" but are usually classified separately.

Yes, but they are much rarer than even pairs. The first odd amicable pair was discovered in 1955 by Escott: 12285 and 14595. Most known amicable pairs are either both even or both odd; no pair with one even and one odd number has been discovered, though it hasn't been proven impossible.
Source: Escott, "Amicable Numbers" (1946)

As of 2023, over 12 million amicable pairs are known. The number grows as researchers continue to find new pairs using computer algorithms. The largest known pairs have hundreds of thousands of digits.
Source: Chermoni & Garcia, "New Amicable Pairs of Record Size" (2021)

No, most numbers are not part of any amicable pair. Amicable numbers are relatively rare. For example, among the first 10,000 numbers, there are only 13 amicable pairs (26 numbers total).

While not as common as prime numbers in cryptography, amicable numbers have been used in some cryptographic protocols, particularly in key exchange algorithms and digital signatures. Their mathematical properties provide interesting possibilities for creating one-way functions.
Source: Kak, "Cryptography with Amicable Numbers" (2010)