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It is relatively straightforward to show that any number '''''P''''' of the form '''''P = 2<sup>n-1</sup> (2<sup>n</sup>-1)''''', for which ('''''2<sup>n</sup>-1''''') is prime, is a perfect number <ref> Proposition IX.36 of Euclid's Elements </ref>. Such numbers are called [[Mersenne]] primes, after Marin Mersenne (1588-1648).
It is relatively straightforward to show that any number '''''P''''' of the form '''''P = 2<sup>n-1</sup> (2<sup>n</sup>-1)''''', for which ('''''2<sup>n</sup>-1''''') is prime, is a perfect number <ref> Proposition IX.36 of Euclid's Elements </ref>. Prime numbers of the form ('''''2<sup>n</sup>-1'''''), for which n is also prime, are called [[Mersenne]] primes, after Marin Mersenne (1588-1648).
The first 12 Mersenne primes are: n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107 and 127; each one of these will generate a perfect number when used in the formula above.
Note that ('''''2<sup>n</sup>-1''''') is not a prime for all prime n, for example ('''''2<sup>11</sup>-1''''') = 2047, which is divisible by 13. The first 12 Mersenne primes are ('''''2<sup>n</sup>-1''''') with: n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107 and 127.
Many of Euclid's successors assumed that all perfect numbers were of this form, [[Euler]] in 1849 provided the first proof that Euclid's formula gives all possible even perfect numbers.
Many of Euclid's successors assumed that all perfect numbers were of this form, [[Euler]] in 1849 provided the first proof that Euclid's formula gives all possible even perfect numbers.