08/02/2021

The perfect numbers and the mysteries they hold.

Imagem: Pixabay

Few people realize it, but the number 6 occupies a place not reserved for just any digit. It belongs to the select group of perfect numbers, distinguished by being equal to the sum of their divisors: 6 can be divided by 1, 2, and 3, which, when added together, result in 6. This discovery is not new. To be more precise, it dates back 2,300 years and is part of one of the oldest and most fascinating branches of mathematics, number theory.

The Greek mathematician Euclid was the first to refer to perfect numbers in his work "The Elements." In the publication, he recounts the discovery of four perfect numbers—6, 28, 496, and 8128—and reveals an effective, but rather time-consuming, way to find others. To give you an idea, it would take more than 1,750 years for another perfect number to be identified.

Before that, Nicomachus of Gerasa, another Greek mathematician, gave a mystical air to the perfect mathematical structure. In his publication "Introduction to Arithmetic," Nicomachus stipulated that if the sum of the divisors of a given number was greater than the number itself, it should be classified as "abundant." Otherwise, if it was a smaller number, it would be "deficient." For some time, his observations left their mark on the field of research and elevated perfect numbers to something divine.

Over the years, mathematicians such as Pietro Antonio Cataldi, Leonhard Euler, and Peter Barlow also made progress in discovering other combinations of digits until we arrived at the total of 51 perfect numbers currently known. One of the curiosities of this group is that they are all even, but it is not yet possible to guarantee that there are no odd perfect numbers. For now, the only thing we know is that, if they exist, they must be greater than 10³⁰⁰, since the conjecture has been computationally verified up to that value.

Numerous minds in the world of mathematics are engaged in the challenging task of making new discoveries in the field of perfect numbers. According to David E. Joyce, Professor of Mathematics and Computer Science at Clark University in the USA, “the traditional criteria of importance in number theory are aesthetic and historical. What people consider important is what interests them,” he stated to the Quora portal.

Although there is no academic consensus on the field, one of the most fascinating things about mathematics is that it reveals wonders that become understandable over time. Who knows what enchantments perfect numbers may reveal in the future? Only time and mathematical advances will tell!

With information from BBC Brazil