Intuition tells mathematicians that adding 2 to a number completely changes its multiplication structure — that is, there is no correlation between whether a number is prime (the multiplication property) and whether the number two units away is prime or not (additional property). Arithmetic theorists have found no evidence that such a correlation exists, but without proof, they cannot rule out the possibility that a correlation may eventually appear.
“For all we know, there may be this vast conspiracy that at a time some n decided to be the head, it has some secret deal with its neighbor n +2 says you’re not allowed to be the leader anymore,” Tao said.
No one can rule out such a conspiracy. That’s why, in 1965, Sarvadaman Chowla came up with a slightly easier way to think about relationships between neighboring numbers. He wanted to show that whether an integer has even or odd primes – a condition known as the “parity” of prime factors – it must not bias its neighboring primes by any means. any way.
This statement is often understood in the sense of the Liouville function, which assigns integers a value of −1 if they have an odd prime (like 12, which is 2 × 2 × 3) and +1 if they have an odd prime. even (like 10, which is equivalent to 2 × 5). The conjecture predicts that there should be no correlation between the values for which the Liouville function takes consecutive numbers.
Many modern methods for studying primes break down when it comes to parity, which is exactly what Chowla surmised. Mathematicians hope that by solving it, they will develop ideas that they can apply to problems like the twin primes conjecture.
For many years, however, it remained nothing more than that: a vain hope. Then, in 2015, everything changed.
Distributed Clusters
Radziwiłł and Kaisa Matomäki of the University of Turku in Finland did not solve the Chowla conjecture. Instead, they wanted to study the behavior of the Liouville function over short time periods. They already know that, on average, the function +1 half the time and -1 half the time. But there is still the possibility that its values might cluster, truncating to long concentrations of all +1 or all −1.
In 2015, Matomäki and Radziwiłł demonstrated that clusters almost never happens. Their work, published the following year, determined that if you pick a random number and look at, say, hundreds or thousands of its nearest neighbors, about half have an integer even and half odd numbers.
“That’s the big piece missing from the puzzle,” said Andrew Granville of the University of Montreal. “They made this unbelievable breakthrough that revolutionized the whole subject.”
It’s strong evidence that numbers aren’t complicit in a large-scale conspiracy — but Chowla’s conjecture is about conspiracies at best. That’s where Tao has come. Within a few months, he figured out how to build on the work of Matomäki and Radziwiłł to attack an easier-to-study version of the problem, the logarithmic Chowla conjecture. In this formula, the smaller numbers are given more weight so that they are likely to be sampled the same way as the larger integers.
Tao had a vision of how a proof of the logarithmic Chowla conjecture could happen. First, he would argue that the logarithmic Chowla conjecture is wrong – that there is in fact a conspiracy between the number of prime factors of consecutive integers. He will then attempt to prove that such a plot can be amplified: An exception to the Chowla conjecture means not only a plot between consecutive integers, but also a negative one. much larger plot along the entire sequence of numbers.
He would then be able to take advantage of the earlier results of Radziwiłł and Matomäki, which correctly ruled out larger plots of this kind. An example of the opposite of the Chowla conjecture would imply a logical contradiction – that is, it cannot exist, and the conjecture must be true.
