239 (number) - in Mathematics

In Mathematics

It is a prime number. The next is 241, with which it forms a pair of twin primes. 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). Because the next odd number, 241 is prime, 239 is a Chen prime.

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/(13)2 is a convergent of the continued fraction of the square root of 2, so that 2392 = 2 · 134 − 1.
• Related to the above, π/4 = 4 arctan(1/5) − arctan(1/239).
• 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.