Prime numbers, arithmetic functions, rational approximations, and Diophantine questions.
19 records · Includes focused interests · How we rate progress
Alpöge and Furman report an unconditional lower bound of 0.6725 for the proportion of nontrivial zeta zeros that are simple and lie on the critical line.
Original report: 2026-08-10Alpöge and Howell report elliptic curves over the rationals with at least 30 and at least 31 independent rational points of infinite order. Dujella’s specialist table records the rank-at-least-31 example as the current record, and the news account attributes assistance to an internal Claude model.
Read original source Original rank-at-least-31 record, point data and attributionelliptic-rank.icarm.cloud
Original report: 2026-08-23Huang, Jackson, Lee, Poonen, Pries and Zhang construct an explicit degree-23 polynomial whose splitting field over the rationals has Galois group M₂₃. They also construct a regular M₂₃ extension of Q(t), completing the realization program for all 26 sporadic finite simple groups.
Read original source Original paper v2: theorems and human/AI attributionarxiv.org
Original report: 2026-08-09OpenAI reports that rational Hodge classes on abelian varieties with good reduction pair rationally with complementary divisor products on the reduction, compatibly across prime-to-p and crystalline realizations, including residue characteristic two.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports the complete Birch–Swinnerton-Dyer leading-term formula for elliptic curves over the rationals whose full q-power Selmer group has corank zero or one at some prime q, including all local prime factors.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports that, for each elliptic curve over the rationals, its quadratic twists of analytic rank zero and one each have density one half among signed squarefree parameters ordered by absolute value.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports cancellation with a power-of-logarithm saving for ordinary two-point Liouville averages along fixed nonproportional affine forms, and a binary corrected Elliott result under uniform nonpretentiousness.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports that no set differing from the positive primes at finitely many integers is a sumset A+B of two nonnegative-integer sets that each contain at least two elements.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports that divergence of the totient-weighted approximation sum gives infinitely many successful shifted approximations for almost every real number, for each fixed real shift and arbitrary finite-valued tolerances.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports at least c_a x/(log x)² primes in each sufficiently large interval (x,2x) for which a fixed integer a is a primitive root, whenever a is neither minus one nor a square.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports modularity of every elliptic curve over every imaginary quadratic number field, with an automorphic representation matching the local parameters at all places.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI’s manuscript reports that all Dirichlet L-functions and finite-order Hecke L-functions over Q(√−3) have no zeros with real part greater than 7/8.
Read original source Original manuscript · family 003github.com
Original report: 2026-10-06The manuscript reports a proof that Catalan’s constant is irrational.
Read original source Original manuscript · family 005github.com
Original report: 2026-10-06The manuscript reports that the irrationality exponent of π equals 2.
Read original source Original manuscript · family 017github.com
Original report: 2026-10-06The manuscript reports h(k) ≪ k²/(log log(3k))², answering the quadratic-bound question in its stated formulation.
Read original source Original manuscript · family 021github.com
Original report: 2026-10-06The manuscript reports that every a/b with 1 ≤ a < b and sufficiently large b has an expansion with O(log log b) distinct positive unit fractions.
Read original source Original manuscript · family 025github.com
Original report: 2026-10-06The manuscript reports a uniform finite bound on component size for every fixed step limit in the Gaussian-prime graph.
Read original source Original manuscript · family 028github.com
Original report: 2026-10-06Anthropic reports an end-to-end Lean proof of Fermat’s Last Theorem and makes the formal development and verification instructions available.
Read original source Original announcementanthropic.com
Original report: 2026-09-04Sothanaphan’s writeup reports infinitely many factorial-divisibility triples with a+b−n in any prescribed positive logarithmic window, while a and b remain bounded away from 0 and n.
Read original source Sothanaphan’s theorem and writeuparxiv.org
Original report: 2026-01-12Prime numbers, arithmetic functions, rational approximations, and Diophantine questions. Follow specific statements rather than treating all number-theoretic progress as a resolution of a famous conjecture.
This topic groups related mathematical questions across publishers. Each focused problem carries its own baseline and each contribution retains its original source. Results are compared only when their assumptions, optimization target, and computation model agree.
Background review: 2026-10-11. Selected sources:
Publication dates and editorial changes.
Claude: a reported improvement in the proportion of zeta zeros on the critical line
Alpöge–Howell–Claude: elliptic curves of rank at least 30 and 31
AI-assisted realization of the Mathieu group M₂₃ over the rationals
OpenAI reports: Artin primitive-root infinitude for every admissible base
OpenAI reports: Weak inhomogeneous Duffin–Schaeffer approximation
Claude: reported end-to-end Lean formalization of Fermat’s Last Theorem
Claude: a reported improvement in the proportion of zeta zeros on the critical line
Erdős #728: an AI-assisted factorial-divisibility proof and human writeup
Claude: reported end-to-end Lean formalization of Fermat’s Last Theorem
Claude: a reported improvement in the proportion of zeta zeros on the critical line
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