Läuchli proved in 1961 that $\mathfrak{n}$ is finite if and only if $2^{2^{\mathfrak{n}}}＜2^{2^{\mathfrak{n}}}\cdot2$. As a consequence, $\mathfrak{n}$ is finite if and only if $2^{2^{2^{\mathfrak{n}}}}＜(2^{2^{2^{\mathfrak{n}}}})^2$. It is still open (asked by Läuchli) whether $\mathfrak{n}$ is finite if and only if $2^{2^{\mathfrak{n}}}＜(2^{2^{\mathfrak{n}}})^2$.

<cite authors="Läuchli, H.">_Läuchli, H._, [**Ein Beitrag zur Kardinalzahlarithmetik ohne Auswahlaxiom**](https://doi.org/10.1002/malq.19610070709), Z. Math. Logik Grundlagen Math. 7, 141-145 (1961). [ZBL0114.01005](https://zbmath.org/?q=an:0114.01005).</cite>

For an English translation of Läuchli's paper, see [here][1].


  [1]: https://drive.google.com/file/d/1sfP2BiCN0VQ77b2_j_XDGZMOSBFpOj-N/view