239 (number)
| ||||
|---|---|---|---|---|
| Cardinal | two hundred thirty-nine | |||
| Ordinal | 239th (two hundred thirty-ninth) | |||
| Factorization | prime | |||
| Prime | yes | |||
| Greek numeral | ΣΛΘ´ | |||
| Roman numeral | CCXXXIX | |||
| Binary | 111011112 | |||
| Ternary | 222123 | |||
| Senary | 10356 | |||
| Octal | 3578 | |||
| Duodecimal | 17B12 | |||
| Hexadecimal | EF16 | |||
239 (two hundred [and] thirty-nine) is the natural number following 238 and preceding 240.
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.[1] It is an Eisenstein prime with no imaginary part and real part of the form 3n − 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 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 − 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.[5]
- The only solutions of the Diophantine equation y2 + 1 = 2x4 in positive integers are (x, y) = (1, 1) or (13, 239).
References
[edit]- ^ Sloane, N. J. A. (ed.). "Sequence A088165 (NSW primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-28.
- ^ "Tables of imaginary quadratic fields with small class number". numbertheory.org.
- ^ "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".
- ^ Weisstein, Eric W. "239". mathworld.wolfram.com. Retrieved 2020-08-20.
- ^ Sloane, N. J. A. (ed.). "Sequence A157017". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.