The Ancient Greeks liked to study the counting numbers 1, 2, 3 etc - which are also called the Natural Numbers. They considered these to be proper numbers - things like 1½, 3.5 and -3 were not considered to be numbers, although the Greeks developed the mathematics for dealing with them. These ancient mathematicians categorised numbers in various ways depending on their properties: even, odd, prime, composite and so on. One of these categories was that of Perfect Numbers.

The study of Perfect Numbers forms part of the branch of mathematics called Number Theory. Some parts of Number Theory have found a use in code-making and breaking, but so far, no-one has found a use for Perfect Numbers. They are pure mathematics, that is, mathematics which is not applied to any real-world situation. Mathematicians study pure mathematics for the enjoyment, and because it can lead into new discoveries. Perfect Numbers are considered interesting by (some) mathematicians because there are unanswered questions about them.

Divisors

In studying the counting numbers, which will just be called numbers for the rest of this Entry, it is found that some numbers divide evenly into others - for example 3 divides evenly into 15. What is meant here is that when you divide 15 by 3, you get a whole number as the result, with nothing left over. 3 is said to be a divisor of 15, because it divides evenly into it.

Prime, Deficient, Abundant and Perfect Numbers

Every number has some divisors. To find out which numbers are divisors of a number N, you can just try every number from one upwards and test it to see if it divides evenly into N. You can stop searching at N/2; a number won't divide evenly into N if it is bigger than half of N. Every number divides evenly into itself, but N is not normally included in the list of divisors of N.

For example, the full list of divisors of 15 is (1, 3, 5).

A number which has only one divisor, the number 1, is called a Prime number. Such numbers are considered to be the building blocks of all numbers, because any number bigger than 1 is either a prime or is made by multiplying primes together.

Inspired by this, the Ancient Greeks thought that the study of divisors was important. They wanted some way of showing that a number had very few divisors, or a lot of divisors. Someone thought up the idea of adding the divisors together. Based on the result, they categorised the numbers into three:

• If the sum of the divisors is less than the number, the number is called deficient.
• If the sum of the divisors is greater than the number, the number is called abundant.
• If the sum of the divisors exactly equals the number, the number is called perfect.

For example, 15 has divisors 1, 3 and 5. If we add these together, we get 1 + 3 + 5 = 9 which is less than 15, so 15 is a deficient number. On the other hand, 12 has divisors 1, 2, 3, 4 and 6. Adding these up gives 1 + 2 + 3 + 4 + 6 = 16 which is greater than 12, so 12 is an abundant number. The number 6 has divisors 1, 2 and 3. Adding these gives 1 + 2 + 3 = 6, so 6 is a perfect number.

Examples of Perfect Numbers

You might think from the way we found a perfect number as small as 6 that perfect numbers are common. In fact they are very rare. It's easy to find that 6 and 28 are both perfect. A bit of searching will eventually show that 496 is also perfect. After that, they are few and far between. The next one is 8,128. The fifth perfect number is a whopping 33,550,336.

To spell this out for 8,128:

• 8,128 has divisors (1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1,016, 2,032, 4,064).
• 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1,016 + 2,032 + 4,064 = 8,128.
• Therefore, 8,128 is perfect.

Even Perfect Numbers

It's easy enough to show using only schoolroom algebra that if M is 1 less than a power of 2 and if M is prime, then N = M*(M+1)/2 is a perfect number.

For example, 2⁷ - 1 = 127 which is prime. Calculating N = 127*128/2 gives 8128, the perfect number that we've already encountered.

It is also possible to show, although the proof is more difficult, that this method of construction gives us all the even perfect numbers. Every even perfect number can be written as M*(M+1)/2, where M = 2^m - 1 and M is prime.

So the search for even perfect numbers equates to the search for primes which are 1 less than a power of 2. Luckily such numbers, known as Mersenne Primes, have been studied in detail and methods are available for searching for them. They are the favourites of computer scientists who want to earn (transient) glory by finding the biggest prime number ever found - it's a good way of showing off the power of their computers. Every time a new Mersenne Prime is found, there is a corresponding perfect number computed.

These numbers are ridiculously big, with thousands of digits in them!

Odd Perfect Numbers

Although the above construction method produces all the even perfect numbers, it says nothing about odd perfect numbers. As yet, nobody knows whether such a thing exists or not. Mathematicians have tested all the odd numbers up as far as one trillion and have shown that none of them are perfect.

While this might seem reasonably good evidence that there is no such number, that's not good enough for mathematicians - they need proof. But so far, all attempts to prove that there are no odd perfect numbers have failed. It's one of the unsolved mysteries of mathematics, although not a particularly important one.

Descartes

The French mathematician and philosopher René Descartes, the man who thought, therefore he was, tackled the problem and showed that if an odd perfect number exists, it must be equal to a prime number multiplied by some number squared¹. That is, it must be of the form p*m² where p is prime.

This means that if you are searching for an odd perfect number, there's no point in looking at numbers which are not in this form. That should narrow down your search a bit.

Euler

The Swiss mathematician Leonhard Euler, who was probably the greatest mathematician ever, also looked at the problem. He managed to prove that the odd perfect number, if it exists, will be expressible in the form i^a.j^b.k^c... where i, j, k... and a are all in the form 4n+1 (that is, one more than a multiple of 4) and b, c... are all even. Descartes's result is actually a special case of this.

Further constraints

Research since Euler has managed to establish some more conditions that the odd perfect number must obey if it exists:

• If divisible by 3, it must have at least 7 other divisors that are prime numbers (known as prime factors).
• If not divisible by 3, it must have at least 11 prime factors.
• Its largest prime factor must be greater than 300,000.
• Its second largest prime factor must be greater than 1,000.
• If it is less than 9,118 digits long, it must have a sixth power of a prime as a divisor.

Again, you would be advised to heed these if you are in the business of searching for that elusive odd perfect number.

Conclusion

So the story is pretty much sewn up for even perfect numbers, but lies wide open for odd ones. If you can find an odd perfect number, or can prove that it is impossible, then you'll get your name written in the mathematical history books!