If as many numbers as we please beginning from a unit be set out continuously in double proportion, until the sum of all becomes prime, and if the sum multiplied into the last make some number, the product will be perfect.
+++++– Euclid, Elements, Book IX.36 (about 300BC)
A perfect number is equal to the sum of its proper divisors, (“proper” means here “not including itself”.) e.g. .
In modern terms, he’s describing the series , and multiplying that by its last term when is prime. i.e.
Primes of the form are now known as Mersenne primes, after their 17th C popularizer.
If is composite, so is . If :
of which is obviously a factor.
But if is prime, isn’t always also prime.
Euler proved that all even perfect numbers are of the form . (No odd perfect numbers have yet been found.)
Every even perfect number is represented in binary as ones followed by zeros:
48 perfect numbers are now known; the most recently found (in 2013) has 34,850,340 digits, and =57,885,161. (see wiki )
William Dunham – Euler: The Master of Us All