# Number theory

> 31 AI-made results in number theory, each explained in plain English with its trust label, Lean status and sources.

Canonical: https://closedform.nishachay.tech/science/mathematics/number-theory/
Last updated: 2026-10-14

31 results in OpenAI Math 2026.

31 AI-made results in number theory, from 58 papers in the OpenAI math collection, each explained in plain English. 17 have a formal Lean proof listed; 14 are claimed in a paper and not yet machine-checked. The oldest question here dates back to 1896: [The quasi-Riemann hypothesis](https://closedform.nishachay.tech/e/openai-math-2026-003/). Results rated Landmark include [The full BSD formula from low Selmer corank](https://closedform.nishachay.tech/e/openai-math-2026-002/) and [Hilbert’s tenth problem over ℚ](https://closedform.nishachay.tech/e/openai-math-2026-004/).

## Results (31)

- [Milne’s rationality conjecture and algebraic specialization](https://closedform.nishachay.tech/e/openai-math-2026-001/) - Entry 001 · After reduction mod p, Hodge classes on abelian varieties are claimed to give one rational number in every cohomology. (Claimed, not checked)
- [The full BSD formula from low Selmer corank](https://closedform.nishachay.tech/e/openai-math-2026-002/) - Entry 002 · The full Birch–Swinnerton-Dyer formula is claimed for every elliptic curve with a small Selmer group at one prime. (Claimed, not checked)
- [The quasi-Riemann hypothesis](https://closedform.nishachay.tech/e/openai-math-2026-003/) - Entry 003 · No zero of Riemann's zeta function, or of any Dirichlet L-function, lies to the right of real part 7/8. (Formal proof listed)
- [Hilbert’s tenth problem over ℚ](https://closedform.nishachay.tech/e/openai-math-2026-004/) - Entry 004 · No algorithm can decide whether a whole-number polynomial equation has a solution in fractions, the paper claims. (Claimed, not checked)
- [Irrationality of Catalan’s constant](https://closedform.nishachay.tech/e/openai-math-2026-005/) - Entry 005 · Catalan's constant, 1 − 1/9 + 1/25 − 1/49 + …, is not a fraction. (Formal proof listed)
- [Goldfeld’s conjecture: densities and mean analytic rank](https://closedform.nishachay.tech/e/openai-math-2026-006/) - Entry 006 · Among the quadratic twists of any elliptic curve, half are claimed to have analytic rank 0 and half rank 1. (Claimed, not checked)
- [Ordinary two-point correlations and the corrected Elliott conjecture](https://closedform.nishachay.tech/e/openai-math-2026-007/) - Entry 007 · Neighboring numbers stop agreeing on prime-factor parity. (Formal proof listed)
- [The Deligne–Drinfeld conjecture](https://closedform.nishachay.tech/e/openai-math-2026-008/) - Entry 008 · A short list of braid equations has exactly the solutions Deligne and Drinfeld predicted, in every size. (Formal proof listed)
- [Function-field reconstruction from Milnor K-theory and Galois data](https://closedform.nishachay.tech/e/openai-math-2026-009/) - Entry 009 · A field of functions can be rebuilt from its multiplication, blurred by powers, plus one trace of addition. (Formal proof listed)
- [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)
- [Restricted geometric Langlands, global Arthur enhancements, and generic Ramanujan](https://closedform.nishachay.tech/e/openai-math-2026-014/) - Entry 014 · Linked papers claim the restricted geometric Langlands equivalence in characteristic p, plus Ramanujan bounds. (Claimed, not checked)
- [Torus-packet equidistribution in prime, quartic, and sextic degrees](https://closedform.nishachay.tech/e/openai-math-2026-015/) - Entry 015 · Number-field grids spread evenly over all grid shapes in prime degrees from five, and in degrees four and six. (Formal proof listed)
- [Zilber–Pink in abelian varieties and the Siegel threefold](https://closedform.nishachay.tech/e/openai-math-2026-016/) - Entry 016 · Unlikely intersections in abelian varieties are claimed to be finite in number, in every dimension. (Claimed, not checked)
- [The irrationality exponent of π is 2](https://closedform.nishachay.tech/e/openai-math-2026-017/) - Entry 017 · Fractions can never get unusually close to π. (Formal proof listed)
- [The Margulis–Platonov conjecture over global fields](https://closedform.nishachay.tech/e/openai-math-2026-018/) - Entry 018 · Normal subgroups of these simple groups over any global field are claimed to come from finitely many local pieces. (Claimed, not checked)
- [The local p -adic section conjecture and global consequences](https://closedform.nishachay.tech/e/openai-math-2026-019/) - Entry 019 · Points on p-adic curves of genus two or more are claimed to match exactly the sections of their fundamental group. (Claimed, not checked)
- [Squarefree quartics and power-free polynomial values](https://closedform.nishachay.tech/e/openai-math-2026-020/) - Entry 020 · Whole-number quartics like n⁴ + 2 take squarefree values at exactly the share the local guess predicts. (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)
- [Short Egyptian fractions](https://closedform.nishachay.tech/e/openai-math-2026-025/) - Entry 025 · Every fraction below 1 is a sum of about log log b distinct unit fractions, as Erdős conjectured. (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)
- [Potential integral density on curve character varieties](https://closedform.nishachay.tech/e/openai-math-2026-027/) - Entry 027 · On every curve, whole-number matrix rules spread across the character variety after one enlargement of the integers. (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)
- [Modularity of elliptic curves over imaginary quadratic fields](https://closedform.nishachay.tech/e/openai-math-2026-030/) - Entry 030 · Every elliptic curve over every imaginary quadratic field is claimed to match an automorphic form, place by place. (Claimed, not checked)
- [Uchida’s conjecture for open homomorphisms of Galois groups](https://closedform.nishachay.tech/e/openai-math-2026-031/) - Entry 031 · Every open map between these Galois groups is claimed to come from a unique map between the fields. (Claimed, not checked)

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