Intuition tells mathematicians that adding 2 to a number should completely change its multiplicative structure – which means there should be no relationship between whether a number is simple (a multiplicative property) and whether the number of two units is simple ( additional property). Number theorists have found no evidence to suggest that such a correlation exists, but without proof, they cannot rule out the possibility that such a correlation may eventually emerge.
“As far as we know, there may be this huge conspiracy that every time a number n decides to be the first, has some secret agreement with his neighbor n + 2, who say you are no longer allowed to be first, “Tao said.
No one has come close to ruling out such a conspiracy. Therefore, in 1965, Sarvadaman Choula formulated a slightly easier way of thinking about the relationship between close numbers. He wanted to show that whether an integer has an even or odd number of prime factors – a condition known as “parity” of the number of its prime factors – must not in any way deviate from the number of prime factors of its neighbors.
This statement is often understood in terms of the Liouville function, which assigns a value of -1 to integers if they have an odd number of prime factors (such as 12, which is equal to 2 × 2 × 3) and +1 if they have an even number. (as 10, which is equal to 2 × 5). The hypothesis assumes that there should be no correlation between the values that the Liouville function takes as consecutive numbers.
Many state-of-the-art methods for studying prime numbers are falling apart when it comes to measuring parity, as Chola suggested. Mathematicians hoped that by solving it, they would develop ideas that could apply to problems such as the hypothesis of simple twins.
For years, however, this remained nothing more than that: fantastic hope. Then, in 2015, everything changed.
Radziwill and Kaisa Matomäki of the University of Turku in Finland did not aim to solve Chola’s conjecture. Instead, they wanted to study the behavior of the Liouville function at short intervals. They already knew that the average function is +1 half time and -1 half time. However, it was possible to group its values, to appear in long concentrations of all +1 or all −1.
In 2015, Matomäki and Radziwiłł proved that these clusters are almost never found. Their work, published the following year, found that if you choose a random number and look at, say, hundreds or thousands of your nearest neighbors, about half have an even number of prime factors and half have an odd number.
“It was the big piece that was missing from the puzzle,” said Andrew Granville of the University of Montreal. “They made this incredible breakthrough that revolutionized the whole subject.”
It was strong evidence that the numbers were not complicit in a large-scale conspiracy – but Chola’s conjecture was for conspiracies at the best level. Tao appeared here. Months later, he saw a way to build on Matomaki and Radziwill’s work to attack a version of the problem that was easier to study, Chola’s logarithmic hypothesis. In this formulation, smaller numbers are given greater weights, so they are just as likely to be sampled as larger integers.
Tao had a vision of how the Chola’s logarithmic hypothesis might prove. First, he would suggest that Chola’s logarithmic hypothesis is incorrect – that there is in fact a conspiracy between the number of prime factors of consecutive integers. He would then try to prove that such a conspiracy could be extended: an exception to Chola’s assumption would mean not just a conspiracy between consecutive integers, but a much larger conspiracy over entire sections of numerical rights.
Then he would be able to take advantage of the previous result of Radziwill and Matomaki, who ruled out larger conspiracies of this kind. A counterexample to Chola’s conjecture would mean a logical contradiction – which means that it cannot exist and the conjecture must be true.