223 – 1 = 8,388,607 = 47 × 178,481Construction work
A combination of Euclid and Euler’s work provides a formula which enables
even perfect numbers to be generated: n is an even perfect number if and only if
n = 2p – 1(2p – 1) where 2p – 1 is a Mersenne prime.
For example, 6 = 2^1 (2^2 – 1), 28 = 2^2 (2^3 – 1) and 496 = 2^4 (2^5 – 1). This
formula for calculating even perfect numbers means we can generate them if we
can find Mersenne primes. The perfect numbers have challenged both people and
machines and will continue to do so in a way which earlier practitioners had not
envisaged. Writing at the beginning of the 19th century, the table maker Peter
Barlow thought that no one would go beyond the calculation of Euler’s perfect
number
230 (2^31 – 1) = 2,305,843,008,139,952,128
as there was little point. He could not foresee the power of modern computers
or mathematicians’ insatiable need to meet new challenges.
Odd perfect numbers
No one knows if an odd perfect number will ever be found. Descartes did not
think so but experts can be wrong. The English mathematician James Joseph
Sylvester declared the existence of an odd perfect number ‘would be little short of
a miracle’ because it would have to satisfy so many conditions. It’s little surprise
Sylvester was dubious. It is one of the oldest problems in mathematics, but if an
odd perfect number does exist quite a lot is already known about it. It would
need to have at least 8 distinct prime divisors, one of which is greater than a
million, while it would have to be at least 300 digits long.