Six is an interesting number. If you add its factors 1, 2 and 3 together, you get 6. This is a neat trick that only works for some numbers, and mathematicians describe these numbers as perfect.
The next perfect number after 6 is 28 (1 + 2 + 4 + 7 + 14 = 28), and the third perfect number is 496. But most numbers aren’t perfect. Numbers with a sum of factors that falls short of the number are known as deficient. An example is 14, where 1 + 2 + 7 = 10. Others, such as 24, have too large a sum, (1 + 2 + 3 + 4 + 6 + 8 + 12 = 36) and are called abundant.
We have found some interesting clues to help us search for perfect numbers. The Greek geometer Euclid found a way of constructing perfect even numbers. He took a type of special number called Mersenne primes and multiplied them by a certain number of 2s. Many years after Euclid, the renowned mathematician Euler showed that all perfect even numbers could be made using Euclid’s formula.
This means Mersenne primes and perfect even numbers come in pairs – every Mersenne prime has a matching perfect number that is even, and vice-versa. Last year, we knew 47 Mersenne primes and 47 perfect even numbers. When scientists discovered a new Mersenne prime earlier this year, they also found the corresponding 48th perfect even number.
We know a lot about perfect even numbers, but we don’t know much about perfect odd numbers. In fact, no one has ever found a perfect number that is odd, and many mathematicians think they don’t exist. But no one has found a proof that these numbers don’t exist. Finding such a number, or the reason why they don’t exist, would solve one of the oldest and most puzzling questions in mathematics.