OpenAI research collection · 3 manuscripts · Source checked 2026-10-19
OpenAI's result summary
Proves that the normalized ordered logarithms of the prime factors of p-1, counted with multiplicity, converge jointly to the Poisson–Dirichlet law PD(1) as p ranges uniformly over primes up to x and x→∞. This resolves the Ford–Konyagin–Luca conjecture. It also proves that infinitely many integers n have more than n^{1-ε} totient preimages, for every ε > 0.
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MANUSCRIPT 01
Weighted dilation graphs, smooth shifted primes and totient fibers
We prove Erdős's conjecture on the largest fibers of Euler's totient function: for every ε > 0, infinitely many positive integers n have more than n^{1-ε } preimages. We also show that, for every fixed δ > 0, there are at least x^{1-o(1)} primes p in 2x< p\le5x whose predecessors have no prime factor exceeding x^(δ).
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For a prime p chosen uniformly from 3≤ p≤ x, list the prime factors of p-1 in decreasing order, with multiplicity. As x→∞, their logarithms, divided by log(p-1), converge in every finite joint distribution to the Poisson–Dirichlet distribution with parameter one. This proves the conjecture of Ford, Konyagin and Luca.
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Prime Predecessors with an Even Number of Prime Factors
We prove that there are infinitely many primes p for which p-1 is squarefree and has an even number of prime factors. Equivalently, there are infinitely many primes p with \mu(p-1)=1.
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New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.