Intuition tells mathematicians that adding 2 to a number should completely change its multiplicative structure - meaning there should be no correlation between whether a number is prime (a multiplicative property) and whether the number two units away is prime (an additive property). “It’s difficult because we are mixing two worlds,” said Oleksiy Klurman of the University of Bristol. The question is challenging because it links two arithmetic operations that usually live independently of one another. For instance, the twin primes conjecture asserts that there are infinitely many primes that differ by only 2 (like 11 and 13). But problems about primes that involve addition have plagued mathematicians for centuries. The primes themselves are defined in terms of multiplication: They’re divisible by no numbers other than themselves and 1, and when multiplied together, they construct the rest of the integers. Many of number theory’s most important problems arise when mathematicians think about how multiplication and addition relate in terms of the prime numbers. Their solution, which pushes techniques from graph theory squarely into the heart of number theory, has reignited hope that the Chowla conjecture will deliver on its promise - ultimately leading mathematicians to the ideas they’ll need to confront some of their most elusive questions. Now, Helfgott and Radziwiłł have provided just that. Mathematicians hoped for a stronger and more widely applicable proof instead. But while the technique he used was heralded as innovative and exciting, it yielded a result that was not precise enough to help make additional headway on related problems, including ones about the primes. It was only a few years ago that mathematicians made any progress, when Tao proved an easier version of the problem called the logarithmic Chowla conjecture. Proving the Chowla conjecture is a “sort of warmup or steppingstone” to answering those more intractable problems, said Terence Tao of the University of California, Los Angeles.Īnd yet for decades, that warmup was a nearly impossible task itself. That seemingly straightforward inquiry is intertwined with some of math’s deepest unsolved questions about the primes themselves.
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