Perfect Numbers and Their Properties

What is a perfect number?

A perfect number is a positive integer that equals the sum of its proper divisors (all divisors except itself).

Step-by-step: Verify that 28 is perfect

Step 1 — Find all divisors of 28: 1, 2, 4, 7, 14, 28.

Step 2 — List the proper divisors (exclude 28 itself): 1, 2, 4, 7, 14.

Step 3 — Sum them:

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

The sum equals the number itself. 28 is perfect! ✓

The known perfect numbers

• $6 = 1 + 2 + 3$
• $28 = 1 + 2 + 4 + 7 + 14$
• $496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248$
• $8128$
• $33550336$

These are extremely rare. As of 2024, only 51 perfect numbers are known, and all of them are even.

The Euclid-Euler connection

Euclid proved: if $2^p - 1$ is prime (a Mersenne prime), then $2^{p-1}(2^p - 1)$ is perfect.

For $p = 2$: $2^2 - 1 = 3$ (prime) → $2^1 \times 3 = 6$ ✓

For $p = 3$: $2^3 - 1 = 7$ (prime) → $2^2 \times 7 = 28$ ✓

For $p = 5$: $2^5 - 1 = 31$ (prime) → $2^4 \times 31 = 496$ ✓

Euler proved the converse: every even perfect number has this form. Whether odd perfect numbers exist is one of the oldest unsolved problems in mathematics.

Abundant and deficient numbers

• Deficient: divisor sum $<$ number (e.g., 8: $1+2+4 = 7 < 8$).
• Perfect: divisor sum $=$ number.
• Abundant: divisor sum $>$ number (e.g., 12: $1+2+3+4+6 = 16 > 12$).

Try it in the visualization

Enter a number. All divisors are found and summed. The result is classified as deficient, perfect, or abundant. An animation shows divisors accumulating toward (or overshooting) the target number.