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Erdős Primitive Sets

A primitive set is a set of integers > 1 in which no element divides another. In 1935, Erdős conjectured that for any such set A, the sum Σ 1/(n·log n) is bounded by the sum over primes. Open for 87 years. Proven by Jared Duker Lichtman in 2022 — by leaning on the von Mangoldt weight.

click to add (or remove) elements from your primitive set:

Your primitive sum

Σ n ∈ A 1 / (n · ln n) = 0
Erdős bound (over primes): 1.6366…
elements
0
ratio to bound
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von Mangoldt weight Λ

The key technical tool. Λ(n) = log p if n = pᵏ for prime p, else 0. Toggle to color cells by Λ — primes glow brightest.
why this matters — and the AI angle
The Erdős primitive sets problem is a textbook case of "easy to state, hard to prove." Pick any primitive set and compute Σ 1/(n log n); the conjecture says you'll never beat the prime-sum ≈ 1.6366. The hard part: there are infinitely many primitive sets to check, including weird ones built from prime powers, square-free semiprimes, and beyond.

Lichtman's 2022 proof turned on the von Mangoldt weight Λ(n) — a function from analytic number theory that "knows" which integers are prime powers. By summing Λ(n)/n² over primitive sets and applying a clever decomposition, Lichtman bounded the original sum. The von Mangoldt re-weighting is the move.

This is the technique recently celebrated in the Sean Eberhard / GPT-5.4 demonstration on Erdős problem #1196: pick the right weighting function from the analytic toolkit and the problem cracks. Not "find a pattern in pretty pictures" — pick up the analytic hammer no one's tried.