The History of Perfect Numbers

The earliest written reference to Perfect Numbers was around 2,500 years ago when the Pythagoreans defined a perfect number as one for whom the sum of its divisors, excluding itself, equals the number itself. For example, 6 can be divided by 1, 2, 3 and 6. Adding these numbers together, excluding 6 itself, gives 6. Therefore 6 is a Perfect Number. Over the centuries, many mathematicians from all over the world have contributed to the finding and defining of perfect numbers.

The First Perfect Numbers

The first Perfect Numbers discovered were 6 and 28. Many early commentators saw religious significance in this. According to Tony Crilly in his book, 50 mathematical ideas you really need to know, published by Quercus in 2007, St Augustine postulated that the perfection of 6 existed before the creation of the world and that the world was created in 6 days because of the perfection of that number. Tobias Dantzig noted in his book, Number: The Language of Science, published by Pi Press in 2005, that others believed the correlation between the Perfect Number 28 and the 28 days of the lunar cycle was a sign of divine intervention.

Nicomachus of Gerasa

By the year 100 the next two perfect numbers, 496 and 8128, had been discovered. The MacTutor History of Mathematics website states that Nichomachus of Gerasa claimed that all Perfect Numbers are even, a theory that is yet to be disproved. He also theorised that there would be one Perfect Number in each range of digits and that they would alternately end in 6 and 8.

With the discovery of more perfect numbers these two theories have been proven incorrect for there are no perfect numbers with 5, 6 or 7 digits. The next to be discovered had 8, followed by a number with 10, and although every perfect number discovered so far has ended in either 6 or 28, they have not alternated between the two.

The Euclid-Euler Theorem

But how have mathematicians been able to discover these perfect numbers? Crilly says that the combined work of two mathematicians, who lived 2000 years apart, provides the answer. Their names were Euclid of Alexandria and Leonhard Euler and their formula, 2^(n−1) X (2n − 1), ^ indicates 'to a power of', known as the Euclid-Euler Theorem, would unlock the secret to generating perfect numbers.

Mersenne Primes

Every good lock must have a key and it was the findings of a French monk named Father Marin Mersenne that would provide the key to Euclid and Euler’s Theorem. Crilly noted that Father Mersenne had worked with numbers generated by 2 to a power, minus 1, or (2^n) -1. These Mersenne numbers include

• 2^1-1=2-1=1
• 2^2-1=4-1=3
• 2^3-1=8-1=7
• 2^4-1=16-1=15
• 2^5-1=32-1=31

Insert a prime Mersenne number into the Euclid-Euler Theorem and the result is a Perfect Number.

Other Facts About Perfect Numbers

With the exception of 6, a perfect number divided by 9 gives remainder 1. All perfect numbers can be arrived at by adding consecutive numbers i.e 1+2+3 … and ending with a Mersenne prime.

By 2007, only 44 perfect numbers were known. The largest took the form of 232,582,656 × (232,582,657 − 1) and consisted of 19,616,714 digits. The hunt continues and mathematicians worldwide are constantly searching for new Mersenne primes with which they can generate even more spectacular perfect numbers. Why don't you visit the GIMPS web page and join in the search?

Or, for some fun math equations check out this article on the Fibonacci Sequence.

Computers Search for Perfect Numbers

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