Prime-Entry Matrices of Fixed Determinant

Definitions and mathematical statement

Fix the matrix norm intended by the source, and let \[\mathcal V=\{A\in M_2({\mathbb{R}}):\det A=2\},\qquad B_T=\{A\in\mathcal V:\|A\|\leq T\}.\] Let $\mathcal P$ be the positive rational primes and define \[N_{\mathrm{prime}}(T)=\#\left\{\begin{pmatrix}a&b\\c&d\end{pmatrix}:a,b,c,d\in\mathcal P,\ ad-bc=2,\ \left\|\begin{pmatrix}a&b\\c&d\end{pmatrix}\right\|\leq T\right\}.\] Entries need not be distinct. The matrices are counted as ordered integral matrices, without quotienting by row/column operations or group actions. The qualitative question is whether infinitely many such matrices exist, equivalently $N_{\mathrm{prime}}(T)\to\infty$ as $T\to\infty$. The wording infinitely many in $B_T$ cannot literally apply to one fixed finite $T$, since a bounded norm ball contains only finitely many integral matrices; it is interpreted in the growing-ball sense explicitly indicated by the quantitative question. The stronger proposed prime matrix theorem is \[\exists C>0:\qquad N_{\mathrm{prime}}(T)\sim C\frac{T^2}{(\log T)^4}\qquad(T\to\infty),\] meaning the ratio tends to one. The norm is not specified, and $C$ is allowed to depend on that fixed norm; no numerical or local-density formula for it is supplied. The determinant is exactly positive two, not its absolute value or a varying determinant. The supplied 2026-08-21 triage labels the question open, says the report explicitly calls it open, and reports no later resolution found.

Short English statement

Are there infinitely many determinant-two integral matrices with four positive prime entries, and does their count in a growing norm ball have the stated $T^2/(\log T)^4$ asymptotic? The source's fixed-ball wording is read explicitly as a limit through growing balls.

Sources

• Local UnsolvedMath registry, permanent ID 30001001, source problem catalogue entry, Prime-Entry Matrices of Fixed Determinant, proposed year 2008; no individual proposer supplied.
• Supplied citation: Real Analysis, Harmonic Analysis and Applications (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/32, pp. 1771--1850: https://doi.org/10.4171/owr/2008/32. Supplied literature label: original Oberwolfach report.