Euclid–Euler theorem

The theorem that an even positive integer is perfect if and only if it equals 2^(p−1)(2^p−1), with 2^p−1 prime.

Perfect number: A positive integer equal to the sum of its positive divisors smaller than itself. The theorem classifies precisely the perfect numbers that are even.
Mersenne number: An integer of the form 2^n−1 for a positive integer n. Every prime candidate in the theorem is a Mersenne number.
Euclid: An ancient Greek mathematician whose surviving work includes the Elements. Book IX of the Elements proves that a Mersenne-prime expression yields an even perfect number.
Odd perfect number: A perfect number that is odd; none is known, and whether any exist remains open. The theorem classifies even perfect numbers but says nothing decisive about odd ones.
Mersenne prime: A prime number of the form 2^p−1, where p is a positive integer. The theorem's formula produces an even perfect number exactly when this factor is prime.
Geometric series: A sum whose successive terms have a constant ratio. Summing the divisors of 2^(p−1) gives the geometric series used in the forward direction.
Elements: Euclid's mathematical treatise presenting geometry and number theory in a systematic form. Its Book IX contains the direction of the theorem later associated with Euclid.
Mersenne prime search: The computational search for prime numbers of the form 2^p−1. Each newly found Mersenne prime immediately yields an even perfect number through the theorem.
Divisor function: An arithmetic function that counts or sums the positive divisors of an integer. The sum-of-divisors function verifies that each number in the theorem's formula is perfect.
Arithmetic function: A function defined on positive integers, often encoding properties of their divisors or prime factors. The divisor-sum identity in the proof is an example of an arithmetic function calculation.

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