Science: Multiperfect numbers proliferate in Colorado

1 May 1993

The largest multiply-perfect numbers are now being found at a prodigious
rate by a mathematician in the US.

A number is said to be perfect if the sum of its proper divisors equals
the number itself. For example, 6 is a perfect number because its proper
divisors are 1, 2 and 3. A number is multiply-perfect if the sum of its
divisors equals an integer multiple of the number. The divisors of 120 add
up to 360, so 120 is a multiply-perfect number with an ‘index’ of 3.

Although perfect numbers and their multiply-perfect cousins serve no
practical purpose, they have amused mathematicians and philosophers from
Euler to Descartes. The ancient Greek mathematician Euclid devised a formula
for perfect numbers that end in 6 or 28. But the less well-behaved multiply-perfect
numbers (or multiperfects, as they are known to aficionados of number theory)
do not follow any known pattern. Theorists must use computers to hunt for
larger numbers.

Now Fred Helenius, a mathematician living in Colorado, has developed
a computer program that is yielding a bumper crop of multiperfects. Since
they were first recognised, only about 700 multiperfects have been found
– the largest of which has an index of 8. But Helenius has increased the
total to 1288, and these include 14 monster multiperfects with indices of
9. The largest, which Helenius discovered two months ago, has 588 digits.


According to Rich Schroeppel, a computer programmer at the University
of Arizona in Tucson who has compiled a list of all known multiperfects,
Helenius’s program is a great improvement on previous algorithms, and is
finding multiperfects at a prodigious rate. ‘Before Fred, we knew of only
a dozen multiperfects of index 8,’ says Schroeppel.

He dubbs Helenius’s algorithm the ‘successive adjustment method’. It
starts with a ‘seed’ number – the product of powers of small primes – and
adjusts the number repeatedly to find a multiperfect.

According to theorists, for each index there appears to be a finite
number of multiperfects. There are only 6 numbers known of index 3, 36 of
index 4, and 65 of index 5. However, there are more than 400 known multiperfects
of index 8, almost all of them found by Helenius. ‘One could say I’ve made
a good start on index 8, and the first touch on the far shore of index 9,’
says Helenius.

But the jury is still out on whether the set of multiperfects is finite,
says Schroeppel. As the numbers get larger, they become more sparsely distributed.
‘The open question is whether they thin out to nothing,’ he says.

Helenius’s holy grail is a multiperfect of index 10, a huge number probably
of more than 700 digits, which may require a supercomputer to calculate
in a reasonable amount of time.

‘I was expecting someone with access to more powerful computing facilities
to take this off my hands,’ says Helenius.