Counting pairs of commuting integer matrices
How many pairs of n×n integer matrices with entries at most T commute? Conjecturally T^{n²+1+ε}; proved for n = 2, 3, open from n = 4.
Cite
@misc{cairn-commuting-integer-matrices-count,
title = {Counting pairs of commuting integer matrices},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/commuting-integer-matrices-count}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY-SA 4.0. Accessed 2026-10-04}
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The problem
The question
Let N_n(T) be the number of pairs (X, Y) of n×n integer matrices with all entries of absolute value at most T and XY = YX. Is it true that for every ε > 0, N_n(T) ≪_ε T^{n²+1+ε}?
What is known
- Lower bound N_n(T) ≫ T^{n²+1}, from taking Y scalar.
- Browning–Sawin–Wang prove N_n(T) ≪ T^{n²+2−2/(n+1)} for all n.
- Follow-up work gives an asymptotic for n = 2 and the expected bound for n = 3, so the first open case is n = 4.
What counts as progress
- The conjectured bound for n = 4, or improved exponents for general n.
- Exact counts for n = 4 and moderate T, split by the dimension of the centraliser of X, to see which strata contribute at order T^{17} — reproducible code and data (level B).
Source. Posed by Tim Browning (with Will Sawin and Victor Wang) in an extended abstract of the Oberwolfach workshop Arithmetic Statistics for Algebraic Objects (2025), recorded in Oberwolfach Reports 52/2025, p. 2780 (EMS Press, DOI 10.4171/OWR/2025/52), licensed under CC BY-SA 4.0. This page summarises the problem in our own words; as an adaptation it is shared under CC BY-SA 4.0 as well.