# 239 (number)

From HandWiki

__: Natural number__

**Short description**
| ||||
---|---|---|---|---|

Cardinal | two hundred thirty-nine | |||

Ordinal | 239th (two hundred thirty-ninth) | |||

Factorization | prime | |||

Prime | yes | |||

Greek numeral | ΣΛΘ´ | |||

Roman numeral | CCXXXIX | |||

Binary | 11101111_{2} | |||

Ternary | 22212_{3} | |||

Quaternary | 3233_{4} | |||

Quinary | 1424_{5} | |||

Senary | 1035_{6} | |||

Octal | 357_{8} | |||

Duodecimal | 17B_{12} | |||

Hexadecimal | EF_{16} | |||

Vigesimal | BJ_{20} | |||

Base 36 | 6N_{36} |

**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; hence, it is also a Chen prime. 239 is a Sophie Germain prime and a Newman–Shanks–Williams prime.^{[1]} It is an Eisenstein prime with no imaginary part and real part of the form 3*n* − 1 (with no exponentiation implied). 239 is also a happy number.

239 is the smallest positive integer *d* such that the imaginary quadratic field **Q**(√−*d*) has class number = 15.^{[2]}

HAKMEM (incidentally AI memo 239 of the MIT AI Lab) included an item on the properties of 239, including these:^{[3]}

- 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.
^{[4]} - 239/169 is a convergent of the continued fraction of the square root of 2, so that 239
^{2}= 2 · 169^{2}− 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
*b*^{n}−*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 2*n*, both included.^{[5]} - The only solutions of the Diophantine equation
*y*^{2}+ 1 = 2*x*^{4}in positive integers are (*x*,*y*) = (1, 1) or (13, 239).

## 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
- 239 is an area code representing part of southwest 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"

## References

- ↑ Sloane, N. J. A., ed. "Sequence A088165 (NSW primes)". OEIS Foundation. https://oeis.org/A088165. Retrieved 2016-05-28.
- ↑ "Tables of imaginary quadratic fields with small class number". http://www.numbertheory.org/classnos/.
- ↑ "Beeler, M., Gosper, R.W., and Schroeppel, R. HAKMEM. MIT AI Memo 239, Feb. 29, 1972. Retyped and converted to html by Henry Baker, April, 1995.". https://www.inwap.com/pdp10/hbaker/hakmem/number.html#item63.
- ↑ Weisstein, Eric W.. "239" (in en). https://mathworld.wolfram.com/239.html.
- ↑ Sloane, N. J. A., ed. "Sequence A157017". OEIS Foundation. https://oeis.org/A157017.

Original source: https://en.wikipedia.org/wiki/239 (number).
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