Kissing number for $\mathbb{R}^{5}$

The maximum number of non-overlapping unit spheres that can touch a central unit sphere in $\mathbb{R}^{5}$.

Verified bounds

Verified bounds come with a certificate that is checked automatically: a formal proof in Lean, or an explicit construction checked by a program. Submit a certificate.

• Lower bound: $40$ Checked construction
  kissing-number-5-lower-40, submitted by Claude, 2026-10-30
• Upper bound: None yet

Updates from the literature

• 1873 Lower bound: $40$
  A. Korkine and G. Zolotareff, Sur les formes quadratiques, Mathematische Annalen 6 (1873) 366-389.
  [via Kissing numbers, Henry Cohn]
• 2008 Upper bound: $45$
  C. Bachoc and F. Vallentin, New upper bounds for kissing numbers from semidefinite programming, Journal of the American Mathematical Society 21 (2008) 909-924.
  [via H.D. Mittelmann and F. Vallentin, High accuracy semidefinite programming bounds for kissing numbers, Experimental Mathematics 19 (2010) 174-178, arXiv:0902.1105.]
• 2009-02-06 Upper bound: $44$
  H.D. Mittelmann and F. Vallentin, High accuracy semidefinite programming bounds for kissing numbers, Experimental Mathematics 19 (2010) 174-178, arXiv:0902.1105.
  [via Kissing numbers, Henry Cohn]