03/04/2024

Leaflet: The Euclid-Euler theorem

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

Around the year 100, the Greek Nicomachus of Gerasa (c. 60–c. 120) published "Introduction to Arithmetic," which became the leading book on the subject for over a thousand years. In it, he formulated five conjectures—statements he believed to be true, but which he did not prove—about perfect numbers.

A number is said to be perfect if it is equal to the sum of its proper divisors, that is, divisors smaller than the number itself. For example, 28 is perfect because its proper divisors are 1, 2, 4, 7, and 14, and the sum of these numbers is 28.

Two of Nicomachus' conjectures were false. Two remain unsolved: they are the oldest unsolved mathematical problems. The other was only proven more than 1,600 years later by Leonhard Euler (1707–1783). That is today's topic.

Four centuries earlier, Euclid had proven that if p is such that 2p ^– 1 is prime, then N = ^2p – 1( ^2p – 1) is perfect. Nicomachus conjectured that every even perfect number is of this form, that is, that Euclid's formula gives all the even perfect numbers that exist.

In 1644, the Frenchman Marin Mersenne (1588–1648) presented a list of eleven values of p such that ^2p – 1 is prime. In fact, he did not check it correctly ("to say whether a 20-digit number is prime or not does not reach the point of time"), so it is clear that the list contained several errors. Still, primes of the form ^2p – 1 are called Mersenne primes.

The first seven numbers in Mersenne's list were already known: the first new example was 2 ³¹ –1 = 2,147,483,647. But that it was actually a prime number was only proven much later by Euler, who in this way found the eighth perfect number: 2 ³⁰ (2 ³¹ –1) = 2,305,843,008,139,952,128. The chronology says a lot about the difficulty of the problem: in 1732, Euler stated that 2 ³¹ –1 is prime, but twenty years later he wrote to his colleague Christian Goldbach that he was unsure about it, and it took him another twenty years to present a proof.

But Euler did even better. In a work carried out around 1747, but only published after his death, he proved Nicomachus' conjecture, which is now called the Euclid-Euler theorem: an even number N is perfect if and only if N = 2p ^-1 (2p ^– 1) for some p such that 2p ^– 1 is prime. Euler is one of the greatest mathematicians in history, perhaps the greatest. But even for him, it must have been cool to share the authorship of a theorem with Euclid, who had lived more than 2,000 years earlier, don't you think?

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