Supercongruences for Domb numbers
Abstract
We prove two supercongruence statements proposed by Z.-W. Sun for Domb numbers. The first concerns the unweighted truncated sum Sp=∑n=0p−1Dn and is governed by the two proper ideal classes of the order of discriminant −60 in ℚ(√(−15)). We prove Sun's conjectured congruences modulo p2 and obtain stronger formulas modulo p3 in the split cases. The second concerns Domb sums weighted by the Lucas sequences associated with X2−11X+1. Sun's original formulation has the exceptional counterexample p=3; after excluding this prime, we prove the corrected statement, including its modulo-p3 refinement when both relevant quadratic characters are +1. Both results arise from one modular mechanism. The twisted Domb series is a modular period on Γ0(6)+⟨ 3⟩ and is related by a quadratic pullback to Zagier's sporadic C-sequence. At primes inert in the imaginary quadratic field underlying the relevant complex multiplication, the truncated C-period is the Hasse invariant and vanishes at supersingular reduction; a finite pullback then gives divisibility modulo p2. At split primes, explicitly constructed determinant-$1$ or determinant-$3$ matrices identify the Beukers unit root, yielding congruences modulo p3. The two applications correspond respectively to orders of complex multiplication of discriminants −60 and −156.
Related
- Finite-sum Identities and Supercongruences for Cubes of Catalan Numbers
- q-Analogues of some supercongruences related to Van Hamme's (E.2) and (F.2) supercongruences
- q-Analogues of some supercongruences related to generalized Van Hamme-type supercongruences
- q-Supercongruences Modulo the Third and Fifth Powers of a Cyclotomic Polynomial
- Proof of Two Conjectural q-Supercongruences Related to Van Hamme's (G.2) Supercongruence
BibTeX
@misc{calderon2026supercongruences,
title = {{Supercongruences for Domb numbers}},
author = {Kevin Calderon},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04772},
eprint = {2610.04772},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04772},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Supercongruences for Domb numbers
AU - Kevin Calderon
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04772
UR - https://arxiv.org/abs/2610.04772
AB - We prove two supercongruence statements proposed by Z.-W. Sun for Domb numbers. The first concerns the unweighted truncated sum S_p=∑_n=0^p−1D_n and is governed by the two proper ideal classes of the order of discriminant −60 in ℚ(√(−15)). We prove Sun's conjectured congruences modulo p^2 and obtain stronger formulas modulo p^3 in the split cases. The second concerns Domb sums weighted by the Lucas sequences associated with X^2−11X+1. Sun's original formulation has the exceptional counterexample p=3; after excluding this prime, we prove the corrected statement, including its modulo-p^3 refinement when both relevant quadratic characters are +1. Both results arise from one modular mechanism. The twisted Domb series is a modular period on Γ_0(6)+⟨ 3⟩ and is related by a quadratic pullback to Zagier's sporadic C-sequence. At primes inert in the imaginary quadratic field underlying the relevant complex multiplication, the truncated C-period is the Hasse invariant and vanishes at supersingular reduction; a finite pullback then gives divisibility modulo p^2. At split primes, explicitly constructed determinant-$1$ or determinant-$3$ matrices identify the Beukers unit root, yielding congruences modulo p^3. The two applications correspond respectively to orders of complex multiplication of discriminants −60 and −156.
ER -