Mathematics.

elementary number theory

Perfect Numbers

Number Theory25 minDifficulty3 out of 10

You should know: divisibility

Overview

A perfect number is a positive integer that equals the sum of its proper divisors (all divisors except itself) — for example 6 = 1 + 2 + 3. The Greeks, especially Euclid in the Elements, studied these numbers for their aesthetic and mystical appeal; Euclid proved that whenever 2ᵖ − 1 is prime (a 'Mersenne prime'), the number 2ᵖ⁻¹(2ᵖ − 1) is perfect. Over two thousand years later, Euler proved the converse: every EVEN perfect number must be of exactly this form, giving a complete classification of even perfect numbers tied one-to-one to Mersenne primes. Whether any odd perfect number exists is one of the oldest open problems in mathematics — none has ever been found, and it is known any odd perfect number (if it exists) would have to be astronomically large and satisfy many restrictive conditions.

Intuition

σ(n) = 2n says the divisors of n split into two equal halves when you count n itself as one of the halves: the proper divisors alone sum to exactly n, a perfect balance. Euclid's construction shows why powers of 2 times a Mersenne prime are naturally 'balanced' this way — the divisors of 2ᵖ⁻¹(2ᵖ−1) are highly structured (powers of 2 times either 1 or the prime factor), and the geometric series 1+2+4+⋯+2ᵖ⁻¹ = 2ᵖ−1 conspires exactly with the Mersenne prime factor to make the proper divisors sum to the number itself.

Formal Definition

Definition

Let σ(n) denote the sum of ALL positive divisors of n (including n itself). Then n is perfect precisely when:

σ(n)=2n\sigma(n) = 2n
Perfect number condition (sum of all divisors = 2n)
2p−1(2p−1) is perfect whenever 2p−1 is prime2^{p-1}(2^{p} - 1) \text{ is perfect whenever } 2^{p} - 1 \text{ is prime}
Euclid's construction (a Mersenne prime 2^p - 1)
Every even perfect number has this form\text{Every even perfect number has this form}
Euler's converse (complete classification of even perfect numbers)

Worked Examples

  1. List the proper divisors of 28 (all divisors except 28 itself).

    1,2,4,7,141, 2, 4, 7, 14

  2. Sum them.

    1+2+4+7+14=281+2+4+7+14 = 28

Answer: 28 is perfect, since its proper divisors sum to 28 itself.

Practice Problems

Difficulty 2/10

Verify that 6 is a perfect number by listing its proper divisors and summing them.

Difficulty 5/10

Use Euclid's formula 2^(p−1)(2^p − 1) with p = 5 to construct a perfect number, given that 2^5 − 1 = 31 is prime.

Difficulty 3/10

Is 12 a perfect number? Check by summing its proper divisors.

Quiz

A perfect number n satisfies:
Euclid's construction 2^(p−1)(2^p − 1) produces a perfect number whenever:
Which of these is still an open problem in number theory?

Summary

  • A perfect number equals the sum of its proper divisors, equivalently σ(n) = 2n.
  • Euclid: 2^(p−1)(2^p − 1) is perfect whenever 2^p − 1 is a Mersenne prime; Euler proved every even perfect number has this form.
  • Whether odd perfect numbers exist at all remains an open problem — none has ever been found.

References

  1. WebsiteWikipedia — Perfect number