Jared Duker Lichtman proved a long-standing conjecture relating prime numbers to a broad class of “primitive” sets. To his adviser, it was a “complete shock.”
, a mathematician at Utrecht University and a former student of Pomerance, essentially “came out of retirement to work with him”—found that a primitive set’s Erdős sum could be no greater than around 1.78. “It’s not too far off,” Martin said. “Only about 10 percent bigger than the conjecture for the primes.”
“It was a variation of Erdős’ original ideas, but it was a very slick, neat way … of getting a not-tight but not-too-bad upper bound,” saidAnd for a few years, that seemed like the best mathematicians could do. It wasn’t clear how to drive that maximum down to 1.64. In the meantime, Lichtman graduated and moved to Oxford to do his doctorate with Maynard, where he’s mainly been working on other problems related to the primes.
But numbers with relatively large prime factors—which are “close” to the primes in some sense—were another story. To deal with them, Lichtman found a way to associate not just one sequence of multiples to each number, but several sequences. As before, the combined density of all those sequences was at most 1. But this time, “these other multiples will kind of grow like weeds and take over some of the space,” Lichtman said.
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