02/02/2022

In Folha, Viana comments on the proof of the 1-3-5 conjecture.

Foto: PxHere

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

Lagrange's four-square theorem states that every positive integer N can be written as the sum of four perfect squares, that is, N = a² + b² + c² + d² for some choice of integers a, b, c, and d.

In general, there are several ways to do this: 310 is equal to both 12+172+42+22 and 92+152+22+02, for example. In 1834, the German Carl Gustav Jacobi (1804–1851) proved a beautiful formula that tells exactly how many ways there are.

In 2016, the Chinese mathematician Zhi-Wei Sun proposed several refinements to the theorem, among which the surprising "1-3-5 conjecture" stands out, according to which it is always possible to choose a, b, c, and c in such a way that a + 3b + 5c is also a perfect square. For example, regarding 310 = 92 + 152 + 22 + 02, we see that 9 + 3 times 15 + 5 times 2 equals 64, which is 82.

Within a year, this statement had already been computationally verified for all integers with fewer than 11 digits. At that point, Sun decided to offer a cash prize for the general proof of the 1-3-5 conjecture, for all integers. The amount was not chosen at random: $1,350.

At the end of 2019, the 1-3-5 conjecture was proven by António Machiavelo, a professor in the mathematics department at the University of Porto, and his doctoral student Nikolaos Tsopanidis, with the help of Rogério Reis, from the computer science department at the same university.

I was particularly pleased because I know both senior authors well: Reis was my classmate in undergraduate studies, we were very close, and Machiavelo, a couple of years younger, was even my student when I taught in Porto, before my doctorate.

The proof of the conjecture has two parts. First, Machiavelli and Tsopanidis proved that the statement is true for all sufficiently large N, say, greater than 105,103,560,127. Then, with the help of Reis, they computationally verified that it is also true for all integers up to that value.

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