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Perfect Number

Perfect Number

A number n for which the sum of all the positive integer factors of n which are less than n add up to n.

For example, 6 and 28 are perfect numbers. The number 6 has factors 1, 2, and 3, and 1 + 2 + 3 = 6. The number 28 has factors 1, 2, 4, 7, and 14, and 1 + 2 + 4 + 7 + 14 = 28.

Key Formula

σ(n)−n=nor equivalentlyσ(n)=2n\sigma(n) - n = n \quad \text{or equivalently} \quad \sigma(n) = 2n
Where:
  • nn = The positive integer being tested
  • σ(n)\sigma(n) = The sum of ALL positive divisors of n, including n itself

Worked Example

Problem: Determine whether 28 is a perfect number.
Step 1: List every positive factor of 28 that is less than 28.
1,  2,  4,  7,  141,\; 2,\; 4,\; 7,\; 14
Step 2: Add those proper divisors together.
1+2+4+7+14=281 + 2 + 4 + 7 + 14 = 28
Step 3: Compare the sum to the original number. The sum equals 28, which is the number itself.
28=28  ✓28 = 28 \; \checkmark
Answer: 28 is a perfect number because the sum of its proper divisors equals 28.

Another Example

Problem: Determine whether 12 is a perfect number.
Step 1: List every positive factor of 12 that is less than 12.
1,  2,  3,  4,  61,\; 2,\; 3,\; 4,\; 6
Step 2: Add those proper divisors together.
1+2+3+4+6=161 + 2 + 3 + 4 + 6 = 16
Step 3: Compare the sum to 12. Since 16 > 12, the sum exceeds the number.
16≠1216 \neq 12
Answer: 12 is not a perfect number. Because the sum of its proper divisors (16) is greater than 12, it is classified as an abundant number.

Frequently Asked Questions

How many perfect numbers are there?
As of now, only 51 perfect numbers have been discovered, and all of them are even. The first four are 6, 28, 496, and 8128. Whether any odd perfect numbers exist is one of the oldest unsolved problems in mathematics — none have ever been found, but no one has proven they cannot exist.
Is there a formula to generate perfect numbers?
Euler proved that every even perfect number has the form 2p−1(2p−1)2^{p-1}(2^p - 1), where 2p−12^p - 1 is a prime (called a Mersenne prime). For example, when p=3p = 3, you get 22×7=282^2 \times 7 = 28. Each new Mersenne prime discovered yields a new even perfect number.

Perfect Number vs. Abundant / Deficient Number

A number is classified by comparing the sum of its proper divisors to itself. If the sum equals the number, it is perfect (e.g., 6). If the sum is greater, it is abundant (e.g., 12, whose proper divisors sum to 16). If the sum is less, it is deficient (e.g., 10, whose proper divisors 1 + 2 + 5 = 8 < 10). Perfect numbers sit exactly on the boundary between abundant and deficient.

Why It Matters

Perfect numbers are among the oldest objects studied in number theory, dating back to Euclid around 300 BCE. Their connection to Mersenne primes links them to modern computational mathematics — finding new Mersenne primes (and thus new perfect numbers) is a major goal of distributed computing projects like GIMPS. They also introduce students to the broader idea of classifying integers by properties of their divisors, a theme that runs throughout algebra and number theory.

Common Mistakes

Mistake: Including the number itself when summing its divisors.
Correction: Only sum the proper divisors — all positive factors less than the number. If you include the number itself, the sum will always be too large. For 6, sum 1 + 2 + 3 = 6, not 1 + 2 + 3 + 6 = 12.
Mistake: Assuming 1 is a perfect number.
Correction: The number 1 has no proper divisors (no positive factors less than 1), so its proper divisor sum is 0, not 1. Therefore 1 is not a perfect number.

Related Terms

  • Factor of an Integer — Proper divisors define perfect numbers
  • Sum — Core operation: summing proper divisors
  • Prime Number — Mersenne primes generate even perfect numbers
  • Integers — Perfect numbers are a subset of positive integers
  • Positive Number — Perfect numbers must be positive
  • Even Number — All known perfect numbers are even
  • Divisor — Alternate term for factor used in this context