Perfect Numbers

It is not known when Perfect Numbers were first discovered, or when they were studied. However, it is thought that they may even have been known to the Egyptians, and may have even been known before. Although the ancient mathematicians knew of the existence of Perfect Numbers, it was the Greeks who took a keen interest in them, especially Pythagoras and his followers (O’Connor and Robertson, 2004).

The Pythagoreans found the number 6 interesting (more for its mystical and numerological properties than for any mathematical significance), as it is the sum of its proper factors, i.e. 6 = 1 + 2 + 3 This is the smallest Perfect Number, the next being 28 (Burton, 1980). These two numbers also had religious significance ascribed to them, as 6 is the number of days it took the Christian God to create the world, and 28 is the number of days in a Lunar Cycle.

ST Augustine even went as far to say Six is a number perfect in itself, and not because god created all things in 6 days; rather the converse is true. God created all things in 6 days because the number is perfect. This he wrote in the City of God (cited in Ellis, 2004).

Though the Pythagoreans were interested in the occult properties of Perfect Numbers, they did little of mathematical significance with them. It was around 300 BC, when Euclid wrote his Elements that the first real result was made. Although Euclid concentrated on Geometry, many number theory results can be found in his text (Burton, 1980).

We shall consider Euclid’s result in a moment, but first, let’s define Perfect Numbers more properly. There are numerous ways to define Perfect Numbers, the early definitions being given in terms of aliquot parts. This author defines Perfect Numbers as: A Perfect Number n, is a positive integer which is equal to the sum of its factors, excluding n itself.

Definition

A Perfect Number N is defined as any positive integer where the sum of its divisors minus the number itself equals the number. The first few of these, already known to the ancient Greeks, are 6, 28, 496, and 8128.

Euclid, over two thousand years ago, showed that all even perfect numbers can be represented by,

N=2p-1(2p -1) where p is a prime for which 2p -1 is a prime number.

That is, we have an even Perfect Number of the form N whenever the Mersenne Number 2p -1 is a prime. Undoubtedly Mersenne was familiar with Euclid’s book in coming up with his primes.

The following gives a table of the first nine Mersenne Primes and Perfect Numbers

Prime, p Mersenne Prime, 2p -1 Perfect Number, 2p-1(2p -1)
2 3 6
3 7 28
5 31 496
7 127 8128
13 8191 33550336
17 131071 8589869056
19 524287 137438691328
31 2147483647 2305843008139952128
61 2305843009213693951 2658455991569831744654692615953842176

Problem: Verify that 28 is a perfect number

Problem 1.2 Verify in the case 18 = 2 · 3 2 = p k q l that the sum σ(n) of all divisors satisfies the formula

\(\sigma (n)\;=(1+P+P^{2}+…P^{k})(1+q+q^{2}+…q^{l})\)


Practise This Question

If two triangles have equal area and stand on the same base, then the altitudes of triangle are equal.