239

Some interesting properties (as well as a few arcane ones not reiterated here) of the number 239 are discussed in Beeler et al. (1972, Item 63). 239 appears in Machin's Formula

$\begin{displaymath} {\textstyle{1\over 4}}\pi = 4\tan^{-1}({\textstyle{1\over 5}}) - \tan^{-1}({\textstyle{1\over 239}}), \end{displaymath}$

which is related to the fact that

$\begin{displaymath} 2 \cdot 13^4 - 1 = 239^2, \end{displaymath}$

which is why 239/169 is the 7th Convergent of $\sqrt{2}\,$ . Another pair of Inverse Tangent Formulas involving 239 is

$\displaystyle \tan^{-1}({\textstyle{1\over 239}})$	$\textstyle =$	$\displaystyle \tan^{-1}({\textstyle{1\over 70}}) - \tan^{-1}({\textstyle{1\over 99}})$
	$\textstyle =$	$\displaystyle \tan^{-1}({\textstyle{1\over 408}}) + \tan^{-1}({\textstyle{1\over 577}}).$

239 needs 4 Squares (the maximum) to express it, 9 Cubes (the maximum, shared only with 23) to express it, and 19 fourth Powers (the maximum) to express it (see Waring's Problem). However, 239 doesn't need the maximum number of fifth Powers (Beeler et al. 1972, Item 63).

References

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972.