Perfect Numbers
Numbers whose sum of proper divisors equals themselves: a millennial mathematical rarity
A perfect number is a natural number that equals the sum of its proper divisors (all its divisors except itself). The simplest example is 6: its proper divisors are 1, 2 and 3, and indeed 1 + 2 + 3 = 6. The next is 28: 1 + 2 + 4 + 7 + 14 = 28. These numbers have fascinated mathematicians for over two thousand years due to their beauty and symmetry.
History of perfect numbers
Perfect numbers were studied by the Pythagoreans in the 6th century BC, who attributed mystical meanings to them and considered them symbols of cosmic harmony. Euclid (300 BC) proved that if 2p − 1 is prime, then 2p−1 × (2p − 1) is a perfect number. Two millennia later, Euler completed the picture by proving that all even perfect numbers have this form. Saint Augustine of Hippo wrote that God created the world in 6 days because 6 is a perfect number, and that the moon orbits the Earth every 28 days for the same reason.
The connection with Mersenne primes
There is a direct correspondence between even perfect numbers and Mersenne primes (primes of the form 2p − 1). Each Mersenne prime generates exactly one even perfect number, and vice versa. For example: 22 − 1 = 3 (prime) produces the perfect number 21 × 3 = 6; 23 − 1 = 7 (prime) produces 22 × 7 = 28; 25 − 1 = 31 (prime) produces 24 × 31 = 496. Finding a new Mersenne prime automatically means discovering a new perfect number.
Open questions
Despite more than two thousand years of study, great mysteries remain unsolved. Are there infinitely many perfect numbers? Most mathematicians believe so, but no one has been able to prove it. Does an odd perfect number exist? It has been shown that if one exists, it must be greater than 101500 and have at least 101 prime factors (not necessarily distinct), but no one has proved they cannot exist. These problems remain open and represent two of the oldest questions in mathematics.
Fascinating properties
Even perfect numbers have curious properties. They all end in 6 or 8 (alternating irregularly). They are all triangular numbers, meaning they can be represented as a triangle of dots. The sum of the reciprocals of the divisors of a perfect number is always exactly 2. Furthermore, every even perfect number (except 6) is the sum of a consecutive series of odd cubes: 28 = 1³ + 3³, 496 = 1³ + 3³ + 5³ + 7³.
Known perfect numbers
To date, 51 perfect numbers are known. The first four are small enough to explore:
Did you know?
- Saint Augustine wrote that God created the world in 6 days because 6 is a perfect number, not the other way around. He also noted the Moon's 28-day cycle — 28 being the second perfect number.
- Every even perfect number is also a triangular number. For example, 6 = T(3) and 28 = T(7), where T(n) = n(n+1)/2.
- The sum of the reciprocals of all divisors of a perfect number always equals 2. For 6: 1/1 + 1/2 + 1/3 + 1/6 = 2. For 28: 1/1 + 1/2 + 1/4 + 1/7 + 1/14 + 1/28 = 2.
- Every even perfect number except 6 can be written as the sum of consecutive odd cubes: 28 = 1³ + 3³, 496 = 1³ + 3³ + 5³ + 7³, 8128 = 1³ + 3³ + 5³ + 7³ + 9³ + 11³ + 13³ + 15³.
- If an odd perfect number exists, it must be greater than 101500 and have at least 101 prime factors. After thousands of years of searching, none has ever been found.
Large perfect numbers
The fifth perfect number is 33,550,336 and the sixth is 8,589,869,056. From there, perfect numbers grow exponentially. The largest known, the 51st perfect number, has more than 49 million digits. The first perfect numbers were discovered by hand by the ancient Greeks, but finding the most recent ones has required supercomputers and months of calculation.
Preguntas Frecuentes
What is a perfect number?
A perfect number is a positive integer that equals the sum of its proper divisors (all divisors except the number itself). The first four perfect numbers are 6 (1+2+3), 28 (1+2+4+7+14), 496, and 8,128. They are extremely rare — only 51 are known to date.
How many perfect numbers are known?
As of 2024, exactly 51 perfect numbers are known. All of them are even. Each corresponds to a Mersenne prime via Euclid's formula: if 2^p − 1 is prime, then 2^(p−1) × (2^p − 1) is a perfect number. The 51st perfect number has over 49 million digits.
Do odd perfect numbers exist?
No odd perfect number has ever been found, and most mathematicians suspect none exists. However, this has not been proven. If an odd perfect number exists, it must be greater than 10^1500, have at least 101 prime factors (not necessarily distinct), and satisfy many other restrictive conditions. This is one of the oldest unsolved problems in mathematics.
Are there infinitely many perfect numbers?
This is unknown. Since every even perfect number corresponds to a Mersenne prime, the question is equivalent to asking whether there are infinitely many Mersenne primes — another open problem. New Mersenne primes (and thus new perfect numbers) are discovered every few years, but a proof of their infinitude remains elusive.
What is the connection between perfect numbers and Mersenne primes?
Euclid proved that if 2^p − 1 is prime (a Mersenne prime), then 2^(p−1) × (2^p − 1) is a perfect number. Euler later proved the converse: every even perfect number has this form. So there is a one-to-one correspondence between Mersenne primes and even perfect numbers. Finding a new Mersenne prime automatically reveals a new perfect number.
Why are perfect numbers called "perfect"?
The ancient Greeks, particularly the Pythagoreans, named them "perfect" (τέλειος, teleios) because they are equal to the sum of their parts — they are neither excessive (abundant) nor deficient. This idea of mathematical harmony and completeness carried philosophical and mystical significance in Greek thought.