In 1770, Joseph-Louis Lagrange proved a beautiful theorem: every positive integer can be written as a sum of four squares, that is, four numbers of the form a^{2} where a is an integer. For example, 7 = 1^{2}+1^{2}+1^{2}+2^{2} (We also know that you can’t write 7 as the sum of less than four squares). The idea of â€‹â€‹the theorem dates back to Diophantus of Alexandria’s “Arithmetic”, written in the 3rd century, and had been explicitly formulated by Claude Bachet in 1621.

But in that same year of 1770 the Englishman Edward Waring (1736â€“1798) was already proposing an even more challenging generalization. In “Analytical Meditations” he stated, without proving: “Every integer is a sum of nine cubes (of the form a^{3}); every integer is also the sum of nineteen bisquares (of the form a^{4})” and added, mysteriously, “and so on.”

Waring was a professor at the University of Cambridge, in England, occupying for almost three decades the position of Lucasian professor, one of the most prestigious in the academic world, with Isaac Newton and Stephen Hawking among its illustrious professors. Today, he is remembered mainly because of the “Meditations”.

In modern parlance, the “Waring problem” is as follows: for every positive integer k there is a number N(k) such that every positive integer can be written as the sum of N(k) powers a^{k} of positive integers? The proof that this is so was only given in 1909 by the German mathematician David Hilbert. And related questions continue to be research topics to this day.

An interesting problem is to explicitly calculate the value of N(k) for each value of k. Lagrange’s four squares theorem means that N(2)=4. The claim that N(3)=9 was proved in 1909 by the Germans Arthur Wieferich and Aubrey Kempner. But N(4)=19 was only proved in 1986 by mathematicians Ramachandran Balasubramanian from India and Jean-Marc Deshouillers and FranÃ§ois Dress from France.

Interestingly, N(5) = 37 came first: it was proved in 1964 by the Chinese mathematician Chen Jingrun. We currently know how to calculate N(k) for every value of k, but some aspects of the formula are still not understood.

One of the most recent and interesting advances in this area took place in the year 2021. It will be the topic of next week.

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