The question is challenging because it links two arithmetic operations that usually live independently of one another. Via QuantaMagazine
whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.
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. But problems about primes that involve addition have plagued mathematicians for centuries. For instance, Many state-of-the-art methods for studying prime numbers break down when it comes to measuring parity, which is precisely what Chowla’s conjecture is all about. Mathematicians hoped that by solving it, they’d develop ideas they could apply to problems like the twin primes conjecture.
He would then be able to take advantage of Radziwiłł and Matomäki’s earlier result, which had ruled out larger conspiracies of exactly this kind. A counterexample to the Chowla conjecture would imply a logical contradiction—meaning it could not exist, and the conjecture had to be true.A Web of Lies To prove his logarithmic version of the Chowla conjecture, Tao needed to show that this graph has too many connections to be a realistic representation of values of the Liouville function. In the language of graph theory, that meant showing that his graph of interconnected numbers had a specific property—that it was an “expander” graph.An expander is an ideal yardstick for measuring the scope of a conspiracy.
But walks on Tao’s graph are strange and circuitous. It’s impossible, for instance, to jump directly from 1,001 to 1,002; that requires at least three steps. A random walk along this graph starts at an integer, adds or subtracts a random prime that divides it, and moves to another integer. Helfgott and Radziwiłł needed to show that this naïve graph approximated Tao’s graph. If they could show that the two graphs were similar, they would be able to infer properties of Tao’s graph by looking at theirs instead. And because they already knew their graph was a local expander, they’d be able to conclude that Tao’s was, too .
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