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Thursday, May 7, 2015

Throwback Fact: Perfect Numbers

Tori's handwriting demonstrates a few properties of the first two perfect numbers
We've mentioned the cult of Pythagoras and his followers a few times in the past. And we know that their mathematical findings did not stop at the Pythagorean theorem. One mathematical idea that fascinated the group was perfect numbers.

A perfect number is defined as a positive integer that is equal to the sum of its positive divisors, excluding itself. The definition appeared as early as Euclid's Elements, and only four of these numbers had been discovered before the Renaissance. The discovery process was so slow because these numbers are inredibly rare- the first is 6. It's a relatively small number, and makes the viewer think that the numbers may common. The next perfect number is 28, and then 496. A Greek mathematician (circa 100 CE) named Nicomachus is credited with the discovery of the fourth perfect number: 8128. Then, more than a millenium passed before an unknown mathematician recorded the fifth perfect number, 33,550,336, for the first time.

Pictured above: Euler (left) and Euclid of the famous theorem

There are a number of very interesting results that come from perfect numbers. When Euclid studied them, he proved that 2p−1(2p − 1) is an even perfect number whenever 2p − 1 is prime (Euclid, Prop. IX.36). About the year 1000, another mathematician conjectured that all even perfect numbers are of that form, and more than 700 years after that, Euler proved that the conjecture is true. The form 2p−1(2p − 1) will always result in an even perfect number. This is known, fittingly, as the Euclid-Euler theorem.

There are many questions that still accompany the study of perfect numbers. The number of perfect numbers that exist is unknown; are there infinitely many? And each of the examples mentioned above is even. Are there any odd perfect numbers? The proofs are left as an exercise to our readers.

1 comment:

  1. The concept of a perfect number is ancient. They've been studied for more than 2200 years, and the newly (January, 2016) discovered perfect number is 274,207,280 x (274,207,281-1). This number is over 44 million digits long. Read more here: