Definition: A pair of different numbers where the sum of the proper divisors of each equals the other.
Chapter 1: Best Friend Numbers (Elementary School Understanding)
Imagine every number has "sharing buddies," which are all the numbers that can be divided into it perfectly (except for itself). We add these buddies up to get a "Friendship Score."
Let's find the Friendship Score for the number 220. Its buddies are {1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110}.
If you add them all up, the Friendship Score is 284.
Now, let's find the Friendship Score for 284. Its buddies are {1, 2, 4, 71, 142}.
If you add them all up, the Friendship Score is 220.
Look at that!
The friendship score of 220 is 284.
The friendship score of 284 is 220.
They are like perfect best friends! Each number's score points directly to the other. A pair of numbers like (220, 284) that are linked in this special way are called amicable numbers. "Amicable" is just a fancy word for friendly. They are a pair of "best friend numbers."
Chapter 2: A Mathematical Friendship (Middle School Understanding)
To understand amicable numbers, we first need to define the sum of proper divisors. The proper divisors of a number n are all of its divisors, excluding n itself. We can write a function for this, s(n).
A pair of different numbers, a and b, are amicable if they form a perfect friendship loop:
The sum of a's proper divisors equals b. (s(a) = b)
The sum of b's proper divisors equals a. (s(b) = a)
The Smallest Amicable Pair: (220, 284)
Let's verify this is a true amicable pair.
For a = 220:
The divisors of 220 are {1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220}.
The proper divisors are all of those except 220.
s(220) = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284.
This matches b, so the first condition is met.
For b = 284:
The divisors of 284 are {1, 2, 4, 71, 142, 284}.
The proper divisors are all of those except 284.
s(284) = 1 + 2 + 4 + 71 + 142 = 220.
This matches a, so the second condition is met.
Since both conditions are met, (220, 284) is a true amicable pair. These numbers are much rarer than perfect numbers. After this first pair was known for centuries, the next pair wasn't discovered until the 1600s by Fermat: (17296, 18416).
Chapter 3: The σ Function and Cycles (High School Understanding)
Amicable numbers are best described using the sum-of-divisors function, σ(n), which is the sum of all divisors of n. The sum of proper divisors, s(n), is simply s(n) = σ(n) - n.
A pair of distinct integers, a and b, is amicable if they satisfy the following system of equations:
σ(a) - a = b
σ(b) - b = a
This can be written more elegantly by adding a to the first equation and b to the second:
σ(a) = a + b
σ(b) = a + b
Therefore, the condition for a pair of numbers (a, b) to be amicable is:
σ(a) = σ(b) = a + b
This reveals the deep structure of the relationship: two numbers are amicable if they have the exact same sum of divisors, and that sum happens to be equal to the sum of the numbers themselves.
Thābit ibn Qurra's Rule:
Around the 9th century, the Arab mathematician Thābit ibn Qurra discovered a rule for generating some amicable pairs.
Let p = 3 × 2ⁿ⁻¹ - 1, q = 3 × 2ⁿ - 1, and r = 9 × 2²ⁿ⁻¹ - 1.
If p, q, and r are all prime numbers for some integer n > 1, then the pair (2ⁿpq, 2ⁿr) is an amicable pair.
For n=2: p=5, q=11, r=71. All are prime. This gives the pair (2² × 5 × 11, 2² × 71) = (220, 284).
For n=4: p=23, q=47, r=1151. All are prime. This gives the pair (17296, 18416).
Chapter 4: Aliquot Sequences and Number Theory (College Level)
Amicable numbers are a special case of a broader concept in number theory: aliquot sequences.
An aliquot sequence is generated by repeatedly applying the sum-of-proper-divisors function, s(n).
n₀, n₁, n₂, ... where n_{k+1} = s(n_k).
The behavior of these sequences is largely an open problem.
Terminating Sequence: The sequence ends at 0 (after reaching 1, since s(1)=0). This is the most common behavior. Example: 10 → s(10)=8 → s(8)=7 → s(7)=1 → 0.
Perfect Number (Cycle of length 1): The sequence becomes stationary at a perfect number. 6 → s(6)=6 → 6....
Amicable Pair (Cycle of length 2): The sequence enters a 2-cycle. 220 → s(220)=284 → s(284)=220 → 284....
Sociable Numbers (Cycle of length > 2): The sequence enters a longer periodic cycle. The smallest is a cycle of length 5 discovered by Poulet: 12496 → 14288 → 15472 → 14536 → 14264 → 12496....
Aspiring Number: A sequence that becomes periodic but does not start with a perfect, amicable, or sociable number.
Untouchable Number: A number that is never the sum of the proper divisors of any other number (i.e., it never appears in an aliquot sequence except possibly as the first term). 5 is an untouchable number.
Open Problems:
Catalan's Aliquot Sequence Conjecture: Conjectures that all aliquot sequences are bounded (i.e., they don't grow to infinity). This is a major unsolved problem. The sequence starting with 276 is the first candidate for an unbounded sequence.
Infinitude of Amicable Pairs: It is unknown whether there are infinitely many amicable pairs. The density of amicable numbers is conjectured to be zero.
Existence of Odd Amicable Pairs: All known amicable pairs have the same parity (both even or both odd). It is unknown if an even-odd amicable pair can exist.
The study of amicable numbers is thus the first step into the deep and largely uncharted territory of iterated number-theoretic functions and their cycles.
Chapter 5: Worksheet - Finding Friends
Part 1: Best Friend Numbers (Elementary Level)
The Friendship Score for 10 is 8 (1+2+5). What is the Friendship Score for 8?
Are 10 and 8 a pair of Best Friend Numbers? Why or why not?
Part 2: Mathematical Friendship (Middle School Level)
Using the s(n) function (sum of proper divisors), show that s(1184) = 1210.
Now calculate s(1210). (The proper divisors are 1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605).
Are (1184, 1210) an amicable pair?
Part 3: The σ Function (High School Level)
Verify that σ(220) = 504 and σ(284) = 504.
Show that 220 + 284 = 504.
Using the σ(a) = σ(b) = a + b definition, explain why this proves (220, 284) is an amicable pair.
A perfect number n can be seen as "amicable with itself." How does the equation s(n) = n fit the amicable pair definition s(a)=b, s(b)=a?
Part 4: Aliquot Sequences (College Level)
What is the definition of a "sociable number"?
Start with the number 12 and write out the first few terms of its aliquot sequence. Does it terminate?
Explain the relationship between perfect numbers, amicable numbers, and sociable numbers using the concept of aliquot sequence cycles.
Thābit ibn Qurra's rule for n=3 gives p=11, q=23, and r=287. Why does this not generate an amicable pair?