Jared Duker Lichtman's side project settles Erdős's primitive set conjecture

Paul Erdős introduced primitive sets in the 1930s: collections of integers greater than 1 in which no member divides another, the primes being the canonical example. He proved that for any such set the sum of 1/(n log n) over its members is always finite — the 'Erdős sum' — and conjectured that the primes, whose sum is about 1.64, attain the maximum. Decades of work confirmed special cases, but the best general upper bound, found in 2019 by Jared Duker Lichtman and his Dartmouth adviser Carl Pomerance, remained near 1.78, and mathematicians described the full conjecture as beyond reach.

Lichtman, then a 26-year-old Oxford PhD student, kept the problem as a side interest since 2018. His insight was to split the argument by prime-factor size: the earlier method pushed small-prime-factor numbers below 1.64, while for numbers with large prime factors he attached several sequences of multiples to each number, letting extras 'grow like weeds and take over space'. The overlapping sequences cut the combined density feeding Mertens' theorem below 1; a precise bound on that drop pushed the worst case under 1.64. He posted the complete proof to arXiv in February 2022.

The reaction mixed astonishment with admiration. His Oxford adviser James Maynard called the sudden completion 'a complete shock'; colleagues noted the proof used none of the heavy machinery the field had been waiting for, only 'some really clever ideas'. Quanta Magazine's profile on 6 June 2022 drew 506 points and 102 comments on Hacker News, and the paper was later published in Forum of Mathematics, Pi (2023). The work cemented the primes as exceptional among primitive sets: their Erdős sum reigns supreme.