A perfect number is a positive integer whose proper divisors — every divisor except the number itself — add up to exactly the number. 6 is the smallest and most famous example: its proper divisors are 1, 2, and 3, and 1+2+3=6.
**Checking a number for perfection.** List every factor of the number except the number itself (the proper divisors), sum them, and compare the sum to the original number. If the sum matches, it's perfect. If the sum is less than the number, the number is called "deficient." If the sum is greater, it's called "abundant."
**Worked example: is 28 perfect?** Factors of 28: 1, 2, 4, 7, 14, 28. Proper divisors (excluding 28 itself): 1, 2, 4, 7, 14. Sum: 1+2+4+7+14 = 28. Yes — 28 is perfect, the second smallest perfect number.
**Worked example: is 12 perfect?** Factors of 12: 1, 2, 3, 4, 6, 12. Proper divisors: 1, 2, 3, 4, 6. Sum: 1+2+3+4+6 = 16. Since 16 > 12, the number 12 is abundant, not perfect — its proper divisors add up to more than the number itself.
**Worked example: is 10 perfect?** Factors of 10: 1, 2, 5, 10. Proper divisors: 1, 2, 5. Sum: 1+2+5 = 8. Since 8 < 10, the number 10 is deficient.
**How rare perfect numbers actually are.** Only four perfect numbers exist below 10,000: 6, 28, 496, and 8,128. The next one, 33,550,336, is dramatically larger — perfect numbers get rare extremely fast as numbers grow. As of the most recent published discoveries, only a few dozen perfect numbers are known in total, despite mathematicians actively searching (aided by computers) for centuries.
**Every known perfect number is even, and there's a reason why.** Euclid proved over two thousand years ago that if 2^p − 1 is a prime number (called a Mersenne prime, after the mathematician who studied numbers of this exact form), then 2^(p−1) × (2^p − 1) is always a perfect number. For p=2: 2² − 1 = 3, which is prime, giving the perfect number 2¹ × 3 = 6. For p=3: 2³ − 1 = 7, prime, giving 2² × 7 = 28. For p=5: 2⁵ − 1 = 31, prime, giving 2⁴ × 31 = 496. Every even perfect number ever found fits this exact pattern (proven by Euler, centuries after Euclid, to be the *only* form even perfect numbers can take).
**No odd perfect number has ever been found — and it's a genuinely open question.** Nobody has ever discovered an odd perfect number, but nobody has proven one can't exist either. It remains one of the oldest unsolved problems in number theory, with mathematicians having ruled out odd perfect numbers below extremely large bounds through computer search, without a general proof either way.
**Why perfect numbers were considered significant historically, beyond pure math.** Ancient Greek mathematicians, including Euclid and later Nicomachus, treated perfect numbers as having a kind of mathematical elegance or even mystical significance — the idea that a number could be exactly reconstructed from its own smaller pieces was seen as a rare and notable property. This significance persisted into later numerology and philosophy, well beyond the number theory itself.
**Common mistakes.** The most frequent error is including the number itself in the sum of divisors — for 6, mistakenly summing 1+2+3+6=12 instead of correctly excluding 6 itself and summing only 1+2+3=6. A second common mistake is confusing "perfect" with simply "has many factors" — abundant numbers (like 12, 18, 20, or 24) have plenty of factors too, but their proper divisors sum to *more* than the number, not exactly equal to it, which is a genuinely different and much more common property.
**Worked example: is 496 perfect?** Factors of 496: 1, 2, 4, 8, 16, 31, 62, 124, 248, 496. Proper divisors (excluding 496): 1, 2, 4, 8, 16, 31, 62, 124, 248. Sum: 1+2+4+8+16+31+62+124+248 = 496. Yes — 496 is perfect, the third known perfect number, confirmed by Euclid's formula: p=5 gives 2⁵−1=31 (prime), and 2⁴×31=16×31=496.
**Worked example: is 20 abundant, deficient, or perfect?** Factors of 20: 1, 2, 4, 5, 10, 20. Proper divisors: 1, 2, 4, 5, 10. Sum: 1+2+4+5+10=22. Since 22>20, the number 20 is abundant.
**Amicable numbers, a related and genuinely fascinating concept worth knowing.** Two numbers are "amicable" if each one's proper divisors sum to the *other* number. The smallest known pair is 220 and 284: the proper divisors of 220 (1,2,4,5,10,11,20,22,44,55,110) sum to 284, and the proper divisors of 284 (1,2,4,71,142) sum to 220. This pair was known to Pythagorean mathematicians in ancient Greece and was considered to have symbolic significance around friendship, given the mutual, reciprocal relationship between the two numbers.
**Worked example: is 33,550,336 perfect, the fifth known perfect number, using Euclid's formula rather than listing every divisor by hand.** p=13: 2¹³−1=8191, which is prime. 2¹²×8191 = 4096×8191 = 33,550,336 — confirmed perfect via the formula, illustrating exactly why perfect numbers get so large so quickly once you move past the first four; listing and summing every proper divisor of a number this size by hand would be impractical, while the formula reaches the answer directly from a single Mersenne-prime check.
**Worked example: is 18 abundant, deficient, or perfect — a quick contrast case.** Factors of 18: 1,2,3,6,9,18. Proper divisors: 1,2,3,6,9. Sum: 1+2+3+6+9=21. Since 21>18, 18 is abundant, joining 12 and 20 as small, common examples of abundant numbers — worth noting that abundant numbers are actually far more common among small integers than perfect numbers are, which is part of why perfect numbers are considered so mathematically special.
**Worked example: is 8,128 perfect, the fourth known perfect number?** Using Euclid's formula with p=7: 2⁷−1=127, which is prime (checked by trial division up to √127≈11.3: not divisible by 2,3,5,7,11). So 2⁶×127 = 64×127 = 8,128 — confirmed perfect by the formula, without needing to sum all of its individual proper divisors by hand.
**Multiply perfect numbers, a genuinely open question worth flagging honestly.** It's not known whether there are infinitely many even perfect numbers, even though every one found so far fits Euclid's pattern tied to Mersenne primes — this remains unproven, tied directly to the still-unresolved question of whether infinitely many Mersenne primes exist at all.
For the underlying factor-finding skill this relies on, see How to Find the Factors of a Number.