two hundred thirty-nine

The MIT Artificial Intelligence Laboratory Memo 239 (entitled HAKMEM) of February 1972 listed various properties of the integer 239, some more interesting than others, such as: that 239 needs the maximum number of powers in Waring’s problem for squares, cubes and fourth powers; that it appears in Machin’s formula for $\pi$:

$$\frac{1}{4}\pi =4{\cot }^{-1}5-{\cot }^{-1}239;$$

that since ${239}^2=2\times {13}^4-1$, the fraction $\frac{239}{{13}^2}$ is a convergent for the continued fraction of $\sqrt{2}$; that since $\pi (1500)=239$ (with $\pi (x)$ being the prime counting function; etc. At the time, the memo couldn’t have mentioned that 239 pounds is Homer Simpson’s weight (as established by at least two episodes of The Simpsons).

The prime 239, like many in its vicinity, is a Chen prime. As a Sophie Germain prime, it begins a Cunningham chain of just length 2 (which ends with 479). On the plane of Eisenstein integers, 239 is an Eisenstein prime (its real part being of the form $3n-1$ and it not having an imaginary part), and it is also a Gaussian prime (its real part being of the form $4n-1$); these two properties it has in common with all real primes $p\equiv 11mod12$. Much more rare is that it is the third Newman-Shanks-Williams prime (the ninth is more than ${10}^{98}$).