# Prime numbers > 12 AI-made results on prime numbers, each explained in plain English. Results about prime numbers: how they are spaced, how they divide other numbers and… Canonical: https://closedform.nishachay.tech/topic/prime-numbers/ Last updated: 2026-10-14 Results about prime numbers: how they are spaced, how they divide other numbers and how they are counted. 12 AI-made results on prime numbers, from 21 papers in the OpenAI math collection, each explained in plain English. 7 have a formal Lean proof listed; 5 are claimed in a paper and not yet machine-checked. They span 2 fields, led by number theory (11) and topology (1). The oldest question here dates back to 1927: [Primitive roots for every admissible integer base](https://closedform.nishachay.tech/e/openai-math-2026-029/). ## Results (12) - [Unrestricted pro-modularity at the prime two](https://closedform.nishachay.tech/e/openai-math-2026-010/) - Entry 010 · Odd two-dimensional 2-adic Galois representations are claimed to come from modular forms, with no condition mod 2. (Claimed, not checked) - [Prime-factor statistics of p-1](https://closedform.nishachay.tech/e/openai-math-2026-011/) - Entry 011 · The prime factors of p − 1, for a random prime p, are claimed to split like a randomly broken stick. (Claimed, not checked) - [Independent largest prime factors of consecutive integers](https://closedform.nishachay.tech/e/openai-math-2026-012/) - Entry 012 · The largest prime factors of n and n + 1 act like independent dice, so each wins half the time. (Formal proof listed) - [Ostmann’s inverse Goldbach conjecture](https://closedform.nishachay.tech/e/openai-math-2026-013/) - Entry 013 · The primes cannot be built by adding two sets of whole numbers. (Formal proof listed) - [A quadratic bound for Jacobsthal’s function](https://closedform.nishachay.tech/e/openai-math-2026-021/) - Entry 021 · Every run of about k² whole numbers holds one sharing no factor with a number that has k prime factors. (Formal proof listed) - [The weak inhomogeneous Duffin–Schaeffer conjecture](https://closedform.nishachay.tech/e/openai-math-2026-022/) - Entry 022 · For any fixed shift, a weighted sum decides whether almost every number is well approximated. (Claimed, not checked) - [Patterson's first moment for cubic Gauss sums](https://closedform.nishachay.tech/e/openai-math-2026-023/) - Entry 023 · Cubic Gauss sums over primes lean positive at exactly the rate Patterson predicted in 1978. (Formal proof listed) - [An asymptotic formula for the number of totients](https://closedform.nishachay.tech/e/openai-math-2026-024/) - Entry 024 · Counting totient values up to x gets an exact leading formula, and doubling x doubles the count in the limit. (Formal proof listed) - [Positive lower density of large prime gaps](https://closedform.nishachay.tech/e/openai-math-2026-026/) - Entry 026 · For any size bar, a fixed share of prime gaps clears it, all the way out. (Formal proof listed) - [Uniformly bounded components of Gaussian-prime graphs](https://closedform.nishachay.tech/e/openai-math-2026-028/) - Entry 028 · No walk through Gaussian primes with steps of bounded length can go on forever. (Formal proof listed) - [Primitive roots for every admissible integer base](https://closedform.nishachay.tech/e/openai-math-2026-029/) - Entry 029 · Every whole number except −1 and perfect squares is claimed to be a primitive root for infinitely many primes. (Claimed, not checked) - [Chromatic splitting: filtrations and counterexamples](https://closedform.nishachay.tech/e/openai-math-2026-318/) - Entry 318 · Five papers claim Hopkins' chromatic splitting conjecture fails in its strong and weak forms, and offer a filtered fix. 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