27/03/2024

In Folha, Marcelo Viana talks about perfect numbers.

Reproduction of Marcelo Viana's column in Folha de S. Paulo.

"Professor, your course started on January 6th and will end on February 28th. Now, 6 is a perfect number, and so is 28! What does that prove about the course?" I heroically resisted answering that my course was perfect, and moved on to discuss the mathematics behind the student's observation.

The concept is quite old: a number is said to be perfect if it is equal to the sum of its proper divisors, that is, the divisors that are smaller than the number itself. For example, 6 is a perfect number because its proper divisors are 1, 2, and 3, and their sum is precisely 6. Similarly, the proper divisors of the number 28 are 1, 2, 4, 7, and 14, and their sum is equal to 28.

The ancient Greeks knew the first four perfect numbers: after 6 and 28 come 496 and 8128. They have intriguing properties. For example, they are all triangular numbers, that is, they are sums of the first integers: 6=1+2+3, 28=1+2+3+4+5+6+7, 496=1+2+3+…+30+31 and 8128=1+2+3+…+126+127.

The first important result about perfect numbers was proven by Euclid , around 300 BC: he showed that if p is a prime number such that 2p ^– 1 is also prime, then N = ^{2p –} ¹ ( ^{2p –} 1) is a perfect number. For p = 2 this gives N = 6, for p = 3 it gives N = 28, for p = 5 it gives N = 496, and for p = 7 it gives N = 8128.

From there it gets more complicated. The next candidate would be p=11, but 2 ¹¹ -1=2047 is not prime, and the corresponding N=2,096,128 is not perfect. In fact, the next perfect numbers would take a millennium and a half to be discovered.

Around 1230, the Egyptian mathematician Ibn Fallus (1194–1252) published a list that, according to him, contained ten perfect numbers. In fact, three were wrong, but he still added three perfect (correct) numbers to the four known to the Greeks: 33,550,336, 8,589,869,056, and 137,438,691,328. However, his work was not disseminated in Europe and was eventually rediscovered during the Italian Renaissance: 33,550,336 appeared in an anonymous manuscript around 1456, and the other two numbers were presented by the Bolognese mathematician Pietro Cataldi (1548–1626) in 1588.

To read the full text, visit the newspaper's website.