239 (number)

239 (two hundred [and] thirty-nine) is the natural number following 238 and preceding 240.

239 is a

• When expressing 239 as a sum of
  cubes (23 is the only other such integer), and the maximum number (19) of fourth powers.^[4]
• 239/
  convergent of the continued fraction of the square root of 2
  , so that 239² = 2 · 169² − 1.
• Related to the above, π/4 rad = 4 arctan(1/5) − arctan(1/239) = 45°.
• 239 · 4649 = 1111111, so 1/239 = 0.0041841 repeating, with period 7.
• 239 can be written as bⁿ − b^m − 1 for b = 2, 3, and 4, a fact evidenced by its
  binary representation 11101111, ternary representation 22212, and quaternary
  representation 3233.
• There are 239 primes < 1500.
• 239 is the largest integer n whose factorial can be written as the product of distinct factors between n + 1 and 2n, both included.^[5]
• The only solutions of the Diophantine equation y² + 1 = 2x⁴ in positive integers are (x, y) = (1, 1) or (13, 239).

References