Adding 2 to a number will change its multiplicative structure, meaning there should be no correlation between the number two units away and the number one units away. Number theorists can't exclude the possibility of a correlation if they don't have a proof.
There could be a conspiracy that every time a number decides to be prime, it has some secret agreement with its neighbor that you are not allowed to be prime anymore.
No one has been able to rule out a conspiracy. That's why the way to think about the relationship between nearby numbers was formulated in 1965, by sarvadaman chowla. He wanted to show that the number of prime factors shouldn't be a factor in determining the number of prime.
The Liouville function assigns a value of 1 if they have an odd number of prime factors and 2 if they do. There should be no correlation between the values that the Liouville function takes.
When it comes to measuring parity, many state-of-the-art methods for studying prime numbers break down. By solving it, mathematicians hoped to develop ideas that could be applied to the twin primes conjecture.
For a long time, it remained a fanciful hope. Everything changed in 2015.
There are clusters that are dispersed.
They didn't set out to solve the Chowla conjecture. They wanted to study the Liouville function over short periods of time. They already knew that, on average, the function is half the time. It was1-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-65561-6556
Those clusters almost never occur, as shown by the results of the research done in 2015. Half of the population has an even number of prime factors and half of the population has an odd number, according to their work.
Andrew Granville of the University of Montreal said that the big piece that was missing from the puzzle was the breakthrough.
It was clear that numbers aren't involved in a large-scale conspiracy, but the Chowla conjecture is about conspiracy at the highest level. That was where Tao came in. He saw a way to attack a version of the problem that is easier to study. Smaller numbers are given larger weights so that they are more likely to be used.
The strategy for using expander graphs to answer a version of the Chowla conjecture was developed by the man. It was courtesy of UCLA.
There was a vision for how a proof of the Chowla conjecture could be done. He would assume that there is a conspiracy between the number of prime factors and the number of consecutive numbers. He would try to show that the conspiracy could be amplified by an exception to the Chowla conjecture.
He would be able to take advantage of the earlier result, which had ruled out larger conspiracies of this kind. A counterexample to the Chowla conjecture would imply a logical contradiction, meaning it could not be.
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