Perfect number

In number theory, a perfect number is a positive integer that equals the sum of its positive proper divisors—that is, all divisors excluding the number itself. For example, the proper divisors of 6 are 1, 2, and 3, and since 1 + 2 + 3 = 6, the number 6 is perfect. Similarly, the proper divisors of 28 sum to 28. The first four perfect numbers known since antiquity are 6, 28, 496, and 8128.
The concept of perfect numbers is one of the oldest topics in number theory. Perfect numbers are intimately linked to the sum-of-divisors function, denoted σ(n). A number n is perfect if
σ(n)=2n,\sigma(n) = 2n,σ(n)=2n,
which is equivalent to its definition in terms of proper divisors. This classical topic continues to inspire mathematical investigation, especially regarding the existence of odd perfect numbers and the infinitude of perfect numbers.

Historical Background

The study of perfect numbers dates back over two millennia. Around 300 BC, Euclid established a significant construction: if 2p−12^{p}-12p−1 is prime, then the number
2p−1(2p−1)2^{p-1}(2^{p}-1)2p−1(2p−1)
is perfect. This result appears in Elements, Book IX, Proposition 36.
By the early Christian era, mathematicians such as Nicomachus of Gerasa were familiar with the first four perfect numbers. Nicomachus described methods equivalent to generating triangular numbers from Mersenne primes, although he did not fully understand that the exponent ppp itself must be prime for the construction to work. He also made the erroneous claim that perfect numbers alternate endings in 6 and 8, which holds only for the first few examples.
Perfect numbers later appeared in philosophical and theological writings. Philo of Alexandria and Augustine of Hippo both connected the perfection of the numbers 6 and 28 with the structure of the natural world. Early medieval Arabic mathematics also contributed: Ibn Fallūs identified several perfect numbers beyond those known to the Greeks, although some he listed were incorrect.
In Renaissance Europe, interest revived with the identification of further perfect numbers in the fifteenth and sixteenth centuries. In 1588, Pietro Cataldi discovered the sixth and seventh perfect numbers and showed that any number produced by Euclid’s rule must end in 6 or 8.
The foundation of the modern theory was laid in the eighteenth century by Leonhard Euler, who proved that every even perfect number must indeed be of Euclid’s form. This result—the Euclid–Euler theorem—establishes a one-to-one correspondence between even perfect numbers and Mersenne primes.

Even Perfect Numbers and Mersenne Primes

A prime of the form 2p−12^{p}-12p−1 is called a Mersenne prime. Euclid’s construction shows that every such prime yields an even perfect number:
Perfect number=2p−1(2p−1).\text{Perfect number} = 2^{p-1}(2^{p}-1).Perfect number=2p−1(2p−1).
The first examples are:

• p=2p = 2p=2: 21(22−1)=2×3=62^{1}(2^{2}-1) = 2 \times 3 = 621(22−1)=2×3=6
• p=3p = 3p=3: 22(23−1)=4×7=282^{2}(2^{3}-1) = 4 \times 7 = 2822(23−1)=4×7=28
• p=5p = 5p=5: 16×31=49616 \times 31 = 49616×31=496
• p=7p = 7p=7: 64×127=812864 \times 127 = 812864×127=8128

Not all numbers of the form 2p−12^{p}-12p−1 with p prime are themselves prime. For example,
211−1=2047=23×89.2^{11}-1 = 2047 = 23 \times 89.211−1=2047=23×89.
Mersenne primes become increasingly rare. Ongoing computational work—especially by the GIMPS (Great Internet Mersenne Prime Search) project—has identified 52 Mersenne primes, and hence 52 even perfect numbers. The largest known perfect number, corresponding to the prime exponent 82,589,933, contains over 82 million digits.
It remains unknown whether infinitely many Mersenne primes exist or whether any odd perfect numbers exist—two longstanding unsolved problems in mathematics.

Special Properties of Even Perfect Numbers

Even perfect numbers have several interesting algebraic and geometric characteristics:

• They are the (2p−1)(2^{p}-1)(2p−1)-th triangular numbers, equal to

  1+2+⋯+(2p−1).1 + 2 + \cdots + (2^{p}-1).1+2+⋯+(2p−1).

• They are the (2p−1)(2^{p}-1)(2p−1)-th hexagonal numbers.
• Except for 6, they are the 2p−13\frac{2^{p}-1}{3}32p−1​-rd centred nonagonal numbers.
• They are equal to the sum of the first 2p−12\frac{2^{p}-1}{2}22p−1​ odd cubes.
• Their digital root is always 1 (except for 6).
• Their binary representation is a sequence of ppp ones followed by p−1p-1p−1 zeros.For example:

  49610=1111100002.496_{10} = 111110000_2.49610​=1111100002​.

• They are pernicious numbers, as their binary representation has a prime number of ones.
• They are practical numbers, meaning they have rich divisor structure.

These properties highlight the deep connections between perfect numbers and other classical number-theoretic sequences.

Odd Perfect Numbers and Open Problems

Despite centuries of research, no odd perfect number has been found. If one exists, it must satisfy many restrictive conditions, such as having at least nine distinct prime factors and being extraordinarily large. Nevertheless, no proof of existence or nonexistence is known.
Likewise, it is unknown whether infinitely many perfect numbers exist. This question is equivalent to the problem of whether infinitely many Mersenne primes exist.
Perfect numbers form part of a wider landscape of divisor-related integer classes, including abundant, deficient, amicable, and sociable numbers. Their study continues to influence algebraic number theory, computational mathematics, and the theory of integer sequences.