Hilbert's tenth problem over the rational numbers
Claims a negative answer: rational solvability of integer polynomial equations (variables unbounded) is undecidable, of Turing degree , and stays so for total degree at most four. The method reduces integer solvability to finitely many rational-solvability tests per input, using rank-one elliptic curves, a five-point height estimate and a parity condition from 2-descent. It does not define existentially in (consistent with Mazur's conjecture), does not bound the number of variables, and does not address other number fields beyond what is known.
Exact FrontierDelta
Scope and record
Occurred: Sep 24, 2026
Delta type: SOURCE CLAIM
Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.
Canonical aliases: Hilbert's tenth problem over the rational numbers · Hilbert's tenth problem over Q
Confidence: Not scored
Registry verification: unreviewed · announcement · candidate
Attribution
VibeMathed
registry · event recorded by
Unreleased internal OpenAI model
model · ai model contributor · OpenAI
Lineage and corrections
Hilbert's tenth problem over the rational numbers parent of this event
Claims a negative answer: rational solvability of integer polynomial equations (variables unbounded) is undecidable, of Turing degree , and stays so for total degree at most four. The method reduces integer solvability to finitely many rational-solvability tests per input, using rank-one elliptic curves, a five-point height estimate and a parity condition from 2-descent. It does not define existentially in (consistent with Mazur's conjecture), does not bound the number of variables, and does not address other number fields beyond what is known. parent of this event
VibeMathed record: Hilbert's tenth problem over the rational numbers evidence for this event
Hilbert's tenth problem over the rational numbers evidence for this event
A pointwise 2-converse for elliptic curves with rational two-torsion (companion, OpenAI, 2026-09-24) evidence for this event
Companion: Fontaine-Mazur modularity at the prime 2 evidence for this event
Hilbert, Mathematische Probleme (1900), Problem 10 evidence for this event
Fontaine-Mazur at p = 2, an input to this proof evidence for this event
Revised edition (October 7, 2026): A pointwise 2-converse for elliptic curves with rational two-torsion evidence for this event
Revised edition (October 6, 2026): Fontaine-Mazur modularity at the prime 2 evidence for this event
Earlier edition (September 24, 2026) evidence for this event
This event attributed to Unreleased internal OpenAI model
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