Perfect numbers

Published juin 14, 2026 · By NumberWiki

A perfect number is exactly equal to the sum of its proper divisors — the parts add back up to the whole. The first is 6 = 1 + 2 + 3. Only 52 perfect numbers have ever been found, the question of whether an odd one exists has been open for more than two thousand years, and every new discovery makes world news in mathematics.

The definition

Take a number and list its proper divisors — every positive divisor except the number itself. Add them up. For most numbers the sum misses: either it falls short (a deficient number — the fate of every prime, whose only proper divisor is 1) or it overshoots (an abundant number, like 12, whose parts 1 + 2 + 3 + 4 + 6 = 16). A perfect number is the knife-edge case where the sum lands exactly:

6 = 1 + 2 + 3
28 = 1 + 2 + 4 + 7 + 14
496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
8128 — the fourth and last that the ancient world knew

After 8128 the gaps become enormous. The fifth perfect number is 33,550,336 — it took until the 15th century to find — and the sixth is 8,589,869,056. The 52nd and largest known has over 49 million digits.

Euclid's machine for making them

Around 300 BCE, Euclid proved a remarkable recipe in the Elements (Book IX, Proposition 36): if 2^p − 1 is prime, then 2^p−1(2^p − 1) is perfect. Try it: for p = 2, 2¹ × 3 = 6. For p = 3, 2² × 7 = 28. For p = 5, 2⁴ × 31 = 496. Each perfect number is a power of two multiplied by a prime that is one less than the next power of two — a prime of the special form we now call a Mersenne prime.

Two thousand years later, Euler closed the loop: every even perfect number has Euclid's form. This Euclid–Euler theorem is one of the longest-gestating collaborations in mathematics — the two halves of the proof are separated by twenty centuries. Its consequence is striking: finding even perfect numbers and finding Mersenne primes are the same problem. Each of the 52 known Mersenne primes yields exactly one perfect number, and vice versa.

The oldest open problem in mathematics

Every known perfect number is even. Does an odd perfect number exist? Nobody knows — and the question is plausibly the oldest unsolved problem in all of mathematics, going back to the Greeks. What we do know is a long list of constraints an odd perfect number would have to satisfy: it must exceed 10¹⁵⁰⁰, have at least 101 prime factors (at least 10 distinct), its largest prime factor must exceed 10⁸, and it must have a very particular algebraic shape worked out by Euler. Carl Pomerance has given a heuristic argument that none should exist; the proofs keep tightening the noose without ever closing it.

A second open question is just as stubborn: are there infinitely many perfect numbers? Equivalent to asking whether there are infinitely many Mersenne primes — conjectured yes, proven nothing.

Numerology, theology, and the name

"Perfect" is not a modern marketing flourish — the name is ancient and came loaded with meaning. The Pythagoreans attached mystical significance to 6 and 28. Early religious commentators noted that creation took 6 days and the lunar month is roughly 28 days; St. Augustine, in The City of God, argued the causality ran the other way — "6 is a perfect number, not because God created all things in six days; rather, God created all things in six days because the number is perfect." Nicomachus of Gerasa (c. 100 CE) catalogued the first four perfect numbers and wrapped them in moral philosophy: perfect numbers were balanced between the "excess" of abundant numbers and the "want" of deficient ones, like virtues between vices.

Nicomachus also guessed, wrongly, that perfect numbers alternate final digits 6, 8, 6, 8 and that there is one per number of digits. Both claims fail — but his catalogue stood as the complete list for thirteen centuries, which may be the longest any wrong conjecture has gone unchallenged for lack of data.

Hunting perfect numbers today

Because of Euclid–Euler, the modern hunt for perfect numbers is the hunt for Mersenne primes, and that hunt is industrialized. The Great Internet Mersenne Prime Search (GIMPS) has coordinated volunteer computers since 1996 and has found every record prime since. Each discovery is verified with the Lucas–Lehmer test — a primality test so efficient for Mersenne candidates that numbers with tens of millions of digits can be certified on commodity hardware. When GIMPS confirms a new Mersenne prime, a new perfect number comes along automatically, free of charge.

Perfect numbers on NumberWiki

All eight perfect numbers that fit in a 64-bit integer are recognized automatically and tagged perfect number; the abundance classification (deficient / perfect / abundant) is computed on every number page from σ(n). Related families to browse: abundant numbers, deficient numbers, semiperfect numbers (a subset of divisors sums to n), and the rare weird numbers (abundant yet not semiperfect). Start with 6, 28, 496, and 8128.