OPEN
This is open, and cannot be resolved with a finite computation.
Let $f(n,k)$ count the number of
self-avoiding walks of $n$ steps (beginning at the origin) in $\mathbb{Z}^k$ (i.e. those walks which do not intersect themselves). Determine\[C_k=\lim_{n\to\infty}f(n,k)^{1/n}.\]
The constant $C_k$ is sometimes known as the connective constant. Hammersley and Morton
[HM54] showed that this limit exists, and it is trivial that $k\leq C_k\leq 2k-1$.
Kesten
[Ke63] proved that $C_k=2k-1-1/2k+O(1/k^2)$, and more precise asymptotics are given by Clisby, Liang, and Slade
[CLS07].
Conway and Guttmann
[CG93] showed that $C_2\geq 2.62$ and Alm
[Al93] showed that $C_2\leq 2.696$. Jacobsen, Scullard, and Guttmann
[JSG16] have computed the first few decimal places of $C_2$, showing that\[C_2 = 2.6381585303279\cdots.\]See also
[529].
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Additional thanks to: Cedric Pilatte
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #528, https://www.erdosproblems.com/528, accessed 2026-01-16
