Erdős Problem #1196
GPT-5.4 Pro produced the key proof idea for a 1968 primitive-sets conjecture on the Erdős sum.
Erdős Problem #1196 asks whether every primitive set A whose elements are all at least x has Erdős sum at most 1 + o(1) as x tends to infinity. In April 2026, Liam Price prompted GPT-5.4 Pro on the problem; Kevin Barreto recognized the result and specialists on the Erdős Problems forum quickly refined and checked the argument. The decisive idea is to keep the problem on the integers: a von Mangoldt-weighted divisibility Markov chain, together with an adjoint hitting-probability bound, controls how often a primitive set can be hit. Tao, Lichtman, Sawin, Barreto, and others then recast the proof using an invariant weight governed by 1/zeta, yielding a clean bound with an explicit error term.
Formula
No element of the set divides another; the primes are the basic example.
The conjecture asks for the asymptotic upper bound of the Erdős sum once all elements of A are large.
The identity sum_{q|n} Lambda(q) = log n makes this a natural downward divisibility chain on the integers.
The zeta-weight reformulation gives an explicit sharpening of the asymptotic result.
Summary
Erdős Problem #1196 asks whether every primitive set A whose elements are all at least x has Erdős sum at most 1 + o(1) as x tends to infinity. In April 2026, Liam Price prompted GPT-5.4 Pro on the problem; Kevin Barreto recognized the result and specialists on the Erdős Problems forum quickly refined and checked the argument. The decisive idea is to keep the problem on the integers: a von Mangoldt-weighted divisibility Markov chain, together with an adjoint hitting-probability bound, controls how often a primitive set can be hit. Tao, Lichtman, Sawin, Barreto, and others then recast the proof using an invariant weight governed by 1/zeta, yielding a clean bound with an explicit error term.