16/10/2024

Folha: 'Addition and multiplication don't get along'

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

Take your favorite integer, n, square it, and add 1. For example, if n=4, the result is ^n² + 1=17. It's a simple calculation, but it holds several mysteries. For example, we don't know if the set of prime numbers in this form ^n² + 1 is finite or infinite.

The problem is that this calculation involves both addition and multiplication, and although these are the two most basic operations in mathematics , the relationship between them is poorly understood. This combination of multiplicative concepts, such as prime numbers, with addition appears in other famous mathematical problems whose solution we also don't know (yet?).

For example, the twin prime problem asks whether there is an infinite set of pairs of primes (such as 3 and 5, or 17 and 19) whose difference is equal to 2. Goldbach's conjecture states that every even integer greater than 4 is the sum of two primes. And the abc conjecture states that if a, b, and c are integers with no common prime factors such that a + b = c, then the product of their prime factors is much greater than c.

In our case, it is clear that when n grows, the result ^n² + 1 also grows. In 1898, the Norwegian Carl Størmer (1874–1957) proved something much more interesting: the largest prime factor of ^n² + 1 also grows arbitrarily (mathematicians say it "goes to infinity") when n grows. But how fast does this happen? This question has kept mathematicians busy for over a century.

The main difficulty is that the behavior of this largest prime factor is very complicated. For example, for n=24,208,143 we have ^n² + 1=586,034,187,508,450, whose largest prime factor is enormous: 67,749,617,053. The next value in the sequence, for n=24,208,144, is ^n² + 1=586,034,235,924,737, whose largest prime factor is only 89.

The first quantitative result was obtained independently in the 1930s by the American Sarvadaman Chowla (1907–1995) and the German Kurt Mahler (1903–1998): they proved that the largest prime factor of n ² +1 is always greater than log (log n), where log represents the natural (Napierian) logarithm.

The sequence log(log n) grows to infinity as n grows, therefore the Chowla–Mahler theorem implies the Størmer theorem. But this growth is very, very slow, and there has always been a belief that the Chowla–Mahler estimate could and should be improved. Despite this, there was almost no progress until the end of 2023, when researcher Hector Pasten , from the Catholic University of Santiago, Chile, achieved the greatest advance on this problem in nine decades.

Pasten has been pondering problems of this kind since 2012, when he began his doctoral studies at Queen's University in Ontario, Canada . In that time, he has become one of the leading experts in this area of number theory. One fine day last November, when he should have been doing something else – preparing an exam for his students – he chose to have some fun thinking about mathematics. I can't cast the first stone…

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