Limits RegistryPUBLIC RECORD
Limits RegistryRoger Apéry proved in 1978 that zeta(3) is irrational, presenting the surprising result at a conference in Marseille to an initially skeptical audience who found his methods opaque. The proof was later clarified and confirmed correct by other mathematicians, including Alfred van der Poorten and Frits Beukers, and it remains the only odd zeta value proven irrational — whether zeta(5), zeta(7), and other odd zeta values are irrational remains open (though Tanguy Rivoal showed in 2000 that infinitely many odd zeta values must be irrational).
The question, scope, and sources behind this Registry record.
Roger Apéry proved in 1978 that zeta(3) is irrational, presenting the surprising result at a conference in Marseille to an initially skeptical audience who found his methods opaque. The proof was later clarified and confirmed correct by other mathematicians, including Alfred van der Poorten and Frits Beukers, and it remains the only odd zeta value proven irrational — whether zeta(5), zeta(7), and other odd zeta values are irrational remains open (though Tanguy Rivoal showed in 2000 that infinitely many odd zeta values must be irrational).
The value zeta(3) = sum over n>=1 of 1/n^3 (approximately 1.2020569..., also called Apéry's constant) is an irrational number.
Change a parameter to stress-test whether a proposed result is still inside the published specification. This is an audit aid, not a proof checker.
This record has no editable parameters. Read the formal question and assumptions before challenging it.
Current frontiers derived from accepted Claims.
An observed or demonstrated result; no opposing bound is implied.
The frontier is not sacred
Most progress starts with a disagreement that survives contact with evidence. If you can push the known lower bound up or pull the upper bound down, show us the work.
≥ when you have shown that at least this value is achievable.≤ when you have shown that anything above this value is impossible.No vibes. State the value, define the scope, and link the paper, proof, code, or reproduction that lets another person check it. Editors review every challenge before the public record changes.
Challenge this recordAssertions tied to evidence, attribution, and review.
The frontier as it changed over time.
Only accepted Claims matching the current specification contribute to the displayed bounds. Strict inequalities remain open; contradictory Claims require editorial review.
1 accepted Claim, with 1 linked evidence records.
Permanent ID limitsregistry.com/limits/LR-APERYS-THEOREM
No active verified bounties are linked to this Limit.
View verified bounty tracker ↗No accepted machine-checked reproductions are recorded for this Limit.
Limits Registry. LR-APERYS-THEOREM. Apéry's Theorem — irrationality of zeta(3). 2026.