Perfect and Amicable Numbers

Key idea

Numbers linked to their divisor sums.

In depth

A number is perfect if σ(n) = 2n. Amicable pairs satisfy that each is the sum of the proper divisors of the other. Euclid showed 2^(p−1)(2^p − 1) is perfect when 2^p − 1 is prime.

Worked example

6 and 28 are perfect. The pair 220 and 284 is amicable.

Real-world uses

• A historical number puzzle linking divisor sums
• A topic in recreational mathematics

Common mistake

Assuming odd perfect numbers have been found; none are known.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Discrete Math & Number Theory

• Strong Induction – Assume all earlier cases to prove the next.
• Well-Ordering Principle – Every nonempty set of positive integers has a least element.
• Partial Orders – Reflexive, antisymmetric, transitive relations.
• Cardinality of Sets – Comparing sizes of infinite sets.
• Countable and Uncountable Sets – ℕ and ℚ are countable; ℝ is not.
• Cantor’s Diagonal Argument – Proof that real numbers are uncountable.