239 (number)

← 238 239 240 →
← 230 231 232 233 234 235 236 237 238 239 → List of numbers ; Integers ; ← 0 100 200 300 400 500 600 700 800 900 →
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

Last updated May 21, 2024

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 − 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.^[5]
The only solutions of the Diophantine equation y² + 1 = 2x⁴ in positive integers are (x, y) = (1, 1) or (13, 239).

Related Research Articles

15 (fifteen) is the natural number following 14 and preceding 16.

21 (twenty-one) is the natural number following 20 and preceding 22.

23 (twenty-three) is the natural number following 22 and preceding 24.

100 or one hundred is the natural number following 99 and preceding 101.

1000 or one thousand is the natural number following 999 and preceding 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000.

300 is the natural number following 299 and preceding 301.

400 is the natural number following 399 and preceding 401.

500 is the natural number following 499 and preceding 501.

700 is the natural number following 699 and preceding 701.

600 is the natural number following 599 and preceding 601.

800 is the natural number following 799 and preceding 801.

900 is the natural number following 899 and preceding 901. It is the square of 30 and the sum of Euler's totient function for the first 54 positive integers. In base 10 it is a Harshad number. It is also the first number to be the square of a sphenic number.

2000 is a natural number following 1999 and preceding 2001.

311 is the natural number following 310 and preceding 312.

167 is the natural number following 166 and preceding 168.

251 is the natural number between 250 and 252. It is also a prime number.

5 (five) is a number, numeral and digit. It is the natural number, and cardinal number, following 4 and preceding 6, and is a prime number. It has garnered attention throughout history in part because distal extremities in humans typically contain five digits.

744 is the natural number following 743 and preceding 745.

30,000 is the natural number that comes after 29,999 and before 30,001.

307 is the natural number following 306 and preceding 308.

References

↑ Sloane, N. J. A. (ed.). "SequenceA088165(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.). "SequenceA157017". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.

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[1] Sloane, N. J. A. (ed.). "SequenceA088165(NSW primes)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-28.

[2] "Tables of imaginary quadratic fields with small class number". numbertheory.org.

[3] "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".

[4] Weisstein, Eric W. "239". mathworld.wolfram.com. Retrieved 2020-08-20.

[5] Sloane, N. J. A. (ed.). "SequenceA157017". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.

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