Erdős Problem #821
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Let \(g(n)\) count the number of \(m\) such that \(\phi(m)=n\).
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Erdős asked whether φ(m) = n can have more than n^(1−ε) solutions for infinitely many n; he proved n^c, Baker–Harman and Lichtman pushed c to 0.7156, and the conjecture C = 1 reduces to a count of primes p with p − 1 having no prime factor above p^δ, a barrier of sieve theory below δ ≈ 0.284. OpenAI's 24 September 2026 paper claims x^(1−o(1)) such primes for every δ, hence the conjecture, in 68 machine-written pages with no Lean for anything but the last page. We read it in ten sectioned hostile seats briefed to refute, checked every cited theorem against its source, had the parity-breaking step's exact identity coded independently twice and made it fail on purpose, and tried nine counterexamples to its shifted-correlation theorem. Verdict on the paper as written: holds as read, ten seats out of ten — no mathematical gap, a sketched bijection, one unwritten inequality, and a theorem that proves more than it claims. Unverified by any human or any kernel.
CONFIRMED