Intuition tells mathematicians that adding 2 to a number should completely change its multiplicative structure – which means that there should be no correlation between whether a number is simple (multiplication property) and whether a two-unit number is simple (additive property). Number theorists have found no evidence to suggest that such a correlation exists, but without evidence they cannot rule out the possibility that it might occur.
“As far as we know, there could be a big conspiracy that counts every time n decides to be prime minister, has some secret deal with a neighbor n + 2 saying that you are no longer allowed to be prime minister, ”Tao said.
No one is even close to ruling out such a conspiracy. So in 1965, Sarvadaman Chowla formulated a slightly easier way of thinking about the relationship between nearby numbers. He wanted to show that whether an integer has an even or odd number of prime factors – a condition known as the “parity” of the number of its prime factors – must in no way diminish the number of prime factors of its neighbors.
This statement is often understood in terms of the Liouville function, which assigns integers a value of -1 if they have an odd number of prime factors (like 12, which is equal to 2 × 2 × 3) and +1 if they have an even number (like 10, which is equal to 2 × 5). The assumption predicts that there should be no correlation between the values that the Liouville function takes for consecutive numbers.
Many state-of-the-art methods for studying prime numbers break down when it comes to measuring parity, which is exactly what Cowla’s assumption says. Mathematicians hoped that by solving this they would develop ideas that could be applied to problems such as the assumption of double primes.
For years, however, nothing more was left than that: imaginative hope. Then, in 2015, everything changed.
Radziwiłł and Kaisa Matomäki from the University of Turku in Finland had no intention of solving Chowlin’s conjecture. Instead, they wanted to study the behavior of the Liouville function at short intervals. They already knew that, on average, the function is +1 half time and -1 half time. But it was still possible that its values could be grouped, erupting in long concentrations of either all +1 or all −1.
In 2015, Matomäki and Radziwiłł proved that these clusters almost never appear. Their work, published the following year, found that if you choose a random number and look at, say, a hundred or thousands of its closest neighbors, about half have an even number of prime factors and half an odd number.
“It was a big piece that the puzzle was missing,” said Andrew Granville of the University of Montreal. “They made this incredible breakthrough that revolutionized the whole subject.”
It was strong evidence that the figures were not complicit in large-scale conspiracies — but Chowla’s speculation about conspiracies is at its finest. That’s where Tao came in. Within a few months, he saw a way to build on Matomäki’s and Radziwiłł’s work to attack a version of the problem that is easier to study, Logarithm’s logarithmic assumption. In this formulation, smaller numbers are given higher weights so they are likely to be sampled as well as larger integers.
Tao had a vision so that he could go to prove the logarithmic Man’s assumption. First, he would assume that Logar’s human logarithmic assumption is false — that there is in fact a conspiracy between the number of prime factors of successive integers. He would then try to show that such a conspiracy could be intensified: an exception to Cowla’s assumption would not only mean a conspiracy between successive integers, but a much larger conspiracy along entire parts of the number line.
He could then take advantage of Radziwiłłł and Matomäki’s earlier result, which ruled out larger conspiracies of this kind. A counterexample to Cowla’s assumption would imply a logical contradiction – meaning it could not exist, and the assumption had to be true.