239 (number)

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

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.
• 239/169 is a 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.

In other fields

239 is also:

• K239 is Mozart's only work for two orchestras.
• 239 is the atomic mass number of the most common isotope of plutonium, Pu-239
• The years A.D. 239 and 239 BC.
• 239 is an area code representing part of Florida in the United States
• 239 is a world famous lyceum of physics in mathematics in Saint-Petersburg, Russia.
• In The Simpsons episode "Homer's Night Out", Homer weighs himself and resolves to exercise. Six months later he weighs himself again and again resolves to exercise. Both times he weighed exactly 239 pounds. His weight was also seen in a daydream in the episode "King-Size Homer"
• 239 is the number of chapters in the Book of Mormon.