Given $d, N \in \mathbb{N}$, we define $\mathfrak{C}_d(N)$ to be the number
of pairs of $d\times d$ matrices $A,B$ with entries in $[-N,N] \cap \mathbb{Z}$
such that $AB = BA$. We prove that $$ N^{10} \ll \mathfrak{C}_3(N) \ll
N^{10},$$ thus confirming a speculation of Browning-Sawin-Wang. We further
establish that $$ \mathfrak{C}_2(N) = K(2N+1)^5 (1 + o(1)),$$ where $K>0$ is an
explicit constant. Our methods are completely elementary and rely on upper
bounds of the correct order for restricted divisor correlations with high
uniformity.

Cet article explore les excursions dans le temps et leurs implications.

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