HAKMEM (incidentally AI memo 239 of theMIT AI Lab) included an item on the properties of 239, including these:[4]
When expressing 239 as a sum ofsquare numbers, 4 squares are required, which is the maximum that any integer can require; it also needs the maximum number (9) of positivecubes (23 is the only other such integer), and the maximum number (19) of fourth powers.[5]
Related to the above,π/4rad = 4 arctan(1/5) − arctan(1/239) = 45°.
239 · 4649 = 1111111, so 1/239 = 0.0041841 repeating, with period 7.
239 can be written asbn − bm − 1 forb = 2, 3, and 4, a fact evidenced by itsbinary representation 11101111,ternary representation 22212, andquaternary representation 3233.
There are 239 primes < 1500.
239 is the largest integern whosefactorial can be written as the product of distinct factors betweenn + 1 and 2n, both included.[6]
The only solutions of the Diophantine equationy2 + 1 = 2x4 in positive integers are (x, y) = (1, 1) or (13, 239).
