One recurrence, two witnesses

The Integer With Nowhere to Stand

Apéry's recurrence manufactures rational approximations to ζ(3) so precise, and with such tightly controlled denominators, that assuming ζ(3) is a fraction forces a positive integer to shrink below one. Run every step.

In June 1978, Roger Apéry presented a sequence that looked implausibly well behaved. The first question was not whether it proved anything. It was whether the recurrence and the sum even made the same numbers.

Press Next. The left route adds binomial terms. The right route knows only the previous two values. Neither route reads the other's result.

The object handed to Marseille

(n+1)³X_n+1 - (34n³ + 51n² + 27n + 5)X_n + n³X_n-1 = 0

The two-route machine

Every integer and fraction below is computed with BigInt arithmetic.

exact arithmetic

Index n: 0

Show exact fractions Show proof obligations Use the 1978 calculator view

Finite binomial sum U_n

computing

sum route

Cubic recurrence U_n

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recurrence route

Finite double sum V_n

computing

sum route

Cubic recurrence V_n

computing

recurrence route

The Marseille ledger

U recurrence residual: computing

V recurrence residual: computing

den(V) divides 2D³: computing

Casoratian: computing

Two roads to ζ(3)

Choose how many terms each road receives. Both bars are rigorous remainder widths, shown on a logarithmic scale. The fast identity is a published identity, not the proof of irrationality.

Terms given to each series: 8

Σ 1/k³

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(5/2) Σ (-1)^k-1 / (k³ C(2k,k))

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Agreement is an exhibit, not yet a proof. Fast approximation alone is never enough. The load-bearing addition is arithmetic: if D_n is the least common multiple of 1 through n, then U_n and 2D_n³V_n are integers.

Fast is not enough

Put a denominator on trial

Suppose, only for the trial, that ζ(3) = p/q for positive whole numbers p and q. The machine searches for an index n where the live linear form

T_n(q) = 2qD_n³ (U_nζ(3) - V_n)

is strictly between zero and one. Under the supposition, the same object must be an integer. Choose q. The numerator p is not needed.

Candidate denominator q

Positive integer, up to 60 digits

q = 0 is excluded. It would make T_n(q) equal zero by definition, a vacuous setting.

Ready for q = 1,000,000

0 < integer < 1

Press Find the trap to compute a certified interval.

The rate budget

The recurrence has dominant root 17 + 12√2. The denominator clearing costs roughly three exponential powers. These are different rates, and the gap between them is what makes the integer trap close.

Dominant root

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17 + 12√2, the larger root of x² - 34x + 1.

Apéry's route

computing

A computed upper bound for the irrationality measure, not its exact value.

Later published bound

5.513891

Rhin and Viola's strict upper bound. This is imported from their theorem.

As of 2026-07-29, individual irrationality of ζ(5) remains open. Fischler's 2026 result proves a collective linear-independence lower bound among odd zeta values. It does not single out ζ(5).

The check

Selected index

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Independent routes

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Denominator clearing

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Exact Casoratian

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Independent ζ(3) bracket

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Positive linear form

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Uncertainties and free choices.

n, q, the series term count, and the three display toggles are free choices made by the reader. The finite-sum bench caps n at 12, the series race at 30 terms, and the denominator field at 60 digits so the page remains responsive.
At n = 0 the page uses D₀ = 1, the empty-lcm convention. Every proof search begins at n = 1.
The page rounds interval endpoints outward. It uses exact BigInt fractions for U, V, D, residuals, divisibility, the Casoratian, and the proof-facing ζ(3) bracket.
The browser checks finitely many instances. The all-n recurrence, integrality, positivity, and decay are imported from the cited proof and its Coq-checked version.
The elementary statement D_n = O(3ⁿ) has an unspecified constant. The interactive search does not use that asymptotic shortcut. It calculates each D_n exactly.
The factors 5/2 and 2D_n³, the recurrence, and 17 + 12√2 are published identities or constants. The 5.513891 bound is displayed from Rhin and Viola, not derived here.
The 13.417820 figure is recomputed from the published rate formula. It and 5.513891 are upper bounds, not measurements and not exact values of the irrationality measure.
For the actual ζ(3), T_n(q) is not claimed to be an integer. Its integrality is conditional on the trial assumption ζ(3) = p/q.
Defining ζ(3) as V_n/U_n would force the error to zero by construction. This page instead brackets ζ(3) with the separate accelerated series.

Reproduce: node research/apery-zeta3/verify-apery-zeta3.mjs

What the HP-67 calculators did, and did not do

Van der Poorten reports that calculators in Marseille evaluated the defining sum and the recurrence separately and kept producing the same values. That numerical agreement made the claim credible enough to pursue. It did not prove the identity for every n. The later telescoping argument, and now the formal verification, supply the missing all-n certificate.

Why the integer is nonzero

The recurrence gives the exact Casoratian V_nU_n-1 - V_n-1U_n = 6/n³. It never vanishes. The rational approximations therefore keep changing. The cited proof turns this identity into an exact positive tail for U_nζ(3) - V_n. The page also certifies positivity independently for every n it displays by enclosing ζ(3) in an alternating-series interval.

What has not been proved here

This establishes irrationality, not transcendence. It gives no closed form for ζ(3), no exact irrationality measure, and no individual result for ζ(5). The accelerated series predates Apéry's proof and is not presented as his discovery.

Sources

Roger Apéry, Irrationalité de ζ2 et ζ3, Astérisque 61 (1979), pp. 11-13. The recurrence and first sequence values appear here.
Alfred van der Poorten, A Proof that Euler Missed (Apéry's Proof of the Irrationality of ζ(3), An Informal Report), The Mathematical Intelligencer 1(4) (1979), pp. 195-203. This is the historical report and source for the denominator factor, Casoratian, and rate bound.
Assia Mahboubi and Thomas Sibut-Pinote, A Formal Proof of the Irrationality of ζ(3), Logical Methods in Computer Science 17(1) (2021). This supplies the complete Coq-checked route and the explicit finite sums.
Georges Rhin and Carlo Viola, The Group Structure for ζ(3), Acta Arithmetica 97(3) (2001), pp. 269-293. The theorem states μ(ζ(3)) < 5.513891.
Stéphane Fischler, Linear Independence of Odd Zeta Values Using Siegel's Lemma, Journal of the London Mathematical Society 113(4) (2026), e70535. It proves a collective result, not individual irrationality of ζ(5).