239 (number)

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

Properties

239 is a prime number. The next is 241, with which it forms a pair of twin primes; hence, it is also a Chen prime. 239 is a Sophie Germain prime and a Newman–Shanks–Williams prime. It is an Eisenstein prime with no imaginary part and real part of the form 3n − 1 (with no exponentiation implied). 239 is a factor of the repdigit 1111111, with the other prime factor being 4649. 239 is also a happy number.

239 is the smallest positive integer d such that the imaginary quadratic field Q() has class number = 15.

239 is the smallest number that contains the highest possible digit in all bases from 2 to 12:

• 11101111₂
• 22212₃
• 3233₄
• 1424₅
• 1035₆
• 461₇
• 357₈
• 285₉
• 239₁₀
• 1A8₁₁
• 17B₁₂

The next number with this property is 5927.

HAKMEM entry

HAKMEM (incidentally AI memo 239 of the MIT AI Lab) included an item on the properties of 239, including these:

• When expressing 239 as a sum of square numbers, 4 squares are required, which is the maximum that any integer can require; it also needs the maximum number (9) of positive cubes (23 is the only other such integer), and the maximum number (19) of fourth powers.
• 239/169 is a convergent of the simple continued fraction of the square root of 2, so that 239² = 2 · 169² − 1.
• Related to the above, = 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&nbsp;+&nbsp;1 and 2n, both included.
• The only solutions of the Diophantine equation y² + 1 = 2x⁴ in positive integers are (x, y) = (1, 1) or (13, 239).

References