239 (number)
239 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. 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.
[HAKMEM] entry
HAKMEM 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 of positive cubes, and the maximum number 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 − bm − 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.
- The only solutions of the Diophantine equation y2 + 1 = 2x4 in positive integers are = or.