Kissing number problem

As of 2026, the exact kissing number $k(n)$ is known only in six dimensions: $k(1) = 2$, $k(2) = 6$, $k(3) = 12$ (Schütte–van der Waerden, 1953), $k(4) = 24$ (Musin, 2003/2008), $k(8) = 240$, and $k(24) = 196,560$ (Odlyzko–Sloane and Levenshtein, 1979). Already in dimension $5$, the problem remains open with $40 \le k(5) \le 44$ (where $40$ is achieved by the root lattice $D_5$), and in dimension $6$ with $72 \le k(6) \le 77$ (where $72$ is achieved by $E_6$). As $n \to \infty$, the best asymptotic lower bound is $k(n) \ge (1+o(1)) \frac{\sqrt{3\pi}}{4\sqrt{2}} \log(3/2) \cdot n^{3/2} (2/\sqrt{3})^n$ (Jenssen, Joos, and Perkins, 2018), while the best asymptotic upper bound remains the 1978 Kabatiansky–Levenshtein bound $k(n) \le 2^{0.401414 n (1 + o(1))}$.