Lagrange’s four squares theorem states that every positive integer N can be written as the sum of four perfect squares, that is, N=atwo+btwo+ctwo+dtwo for some choice of integers a, b, c, and d.
In general, there are several ways to do this: 310 is equal to either 1two+17two+4two+2two as for 9two+15two+2two+0two, for example. In 1834, the German Carl Gustav Jacobi (1804–1851) proved a beautiful formula that tells you exactly how many ways there are.
In 2016, Chinese mathematician Zhi-Wei Sun proposed several refinements of 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 d in such a way that a+ 3b+5c is also a perfect square. For example, relative to 310=9two+15two+2two+0two we see that 9 + 3 times 15 + 5 times 2 is equal to 64, which is 8two.
Within a year, this claim had already been computationally verified for all integers with less than 11 digits. At this 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 dollars.
At the end of 2019, the 1-3-5 conjecture was proved by António Machiavelo, 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 department at the same university.
I was particularly happy because I know the two senior authors well: Reis was my undergraduate colleague, we were very close, and Machiavelo, a couple of years younger, became my student when I taught in Porto, before my doctorate.
The proof of the conjecture has two parts. First, Machiavelo and Tsopanidis proved that the statement is true for every 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.
A curious aspect of the proof is that it is based on quaternions, a peculiar type of numbers discovered in 1843 by the Irishman William Hamilton (1805–1865). Initially, quaternions would be used in geometry and mechanics, but they also appear here in an important role in number theory.
Source: Folha
