A note on a recent claimed proof of the irrationality of Catalan's constant
Abstract
The preprint arXiv:2609.04176 (Z.-W. Sun, 3 September 2026) claims a proof that Catalan's constant G is irrational: it constructs for each B a number q̂B from weighted tails of the series and asserts that, if G=a/q, the integer NB=qS HBminq̂B satisfies 0<|NB|<1 for large B (its Theorem 9.1). We report four exact computations. First, the printed proof of the rank theorem (Theorem 2.1) carries Ti where its own recurrence requires Ti+1; the slip is repairable, and the statement is verified exactly at 44 parameter pairs with S≤ 4, B≤ 30. Second, with the paper's definitions and the worst-case denominator, attained at an explicit rational a/q, a lower bound for the quantity the derivation of Theorem 9.1 controls equals +1.49 to +1.79 B2 at eighteen indices 20≤ B≤ 119, where the derivation claims at most −0.00966 B2+o(B2). Third, the paper's own denominator bound (Corollary 5.2) holds in every tested instance and is nearly sharp, so the discrepancy lies in the asymptotic ledger of Sections 6-9. Fourth, an exact identity locates it: the prime 2 contributes v2(FD)log 2=(2log 2+o(1))B2 to the quantity, because the positive part at 2 vanishes while |q̂B| carries the full power of 2 in FD; no display in the ledger carries a term of this order, and Remark 9.3 assigns it to the derivation of the odd-prime constant codd=0.006 without showing how. At the tested parameters the odd-prime layers agree with the paper's constants to within about 0.1 B2, and the remainder of the discrepancy is the size of the residual minor at those indices. These computations test the paper's bounding method, which is uniform in (a,q); they show that the published derivation does not justify the claimed estimate, so the proof of the main theorem is incomplete. Nothing is concluded about the irrationality of G.
Related
BibTeX
@misc{wachs2026note,
title = {{A note on a recent claimed proof of the irrationality of Catalan's constant}},
author = {Dane Wachs},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22339},
eprint = {2609.22339},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22339},
}
RIS
TY - UNPB
TI - A note on a recent claimed proof of the irrationality of Catalan's constant
AU - Dane Wachs
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.22339
UR - https://arxiv.org/abs/2609.22339
AB - The preprint arXiv:2609.04176 (Z.-W. Sun, 3 September 2026) claims a proof that Catalan's constant G is irrational: it constructs for each B a number q̂_B from weighted tails of the series and asserts that, if G=a/q, the integer N_B=q^S H_B^minq̂_B satisfies 0<|N_B|<1 for large B (its Theorem 9.1). We report four exact computations. First, the printed proof of the rank theorem (Theorem 2.1) carries T_i where its own recurrence requires T_i+1; the slip is repairable, and the statement is verified exactly at 44 parameter pairs with S≤ 4, B≤ 30. Second, with the paper's definitions and the worst-case denominator, attained at an explicit rational a/q, a lower bound for the quantity the derivation of Theorem 9.1 controls equals +1.49 to +1.79 B^2 at eighteen indices 20≤ B≤ 119, where the derivation claims at most −0.00966 B^2+o(B^2). Third, the paper's own denominator bound (Corollary 5.2) holds in every tested instance and is nearly sharp, so the discrepancy lies in the asymptotic ledger of Sections 6-9. Fourth, an exact identity locates it: the prime 2 contributes v_2(F_D)log 2=(2log 2+o(1))B^2 to the quantity, because the positive part at 2 vanishes while |q̂_B| carries the full power of 2 in F_D; no display in the ledger carries a term of this order, and Remark 9.3 assigns it to the derivation of the odd-prime constant c_odd=0.006 without showing how. At the tested parameters the odd-prime layers agree with the paper's constants to within about 0.1 B^2, and the remainder of the discrepancy is the size of the residual minor at those indices. These computations test the paper's bounding method, which is uniform in (a,q); they show that the published derivation does not justify the claimed estimate, so the proof of the main theorem is incomplete. Nothing is concluded about the irrationality of G.
ER -