239 (number)

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

← 238 239 240 →
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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
Quaternary	32334
Quinary	14245
Senary	10356
Octal	3578
Duodecimal	17B12
Hexadecimal	EF16
Vigesimal	BJ20
Base 36	6N36

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 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.
• 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.[4]
• The only solutions of the Diophantine equation y² + 1 = 2x⁴ 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
• The years A.D. 239 and 239 BC.
• 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"
• 239 is the number of chapters in the Book of Mormon.

References

1. Sloane, N. J. A. (ed.). "Sequence A088165 (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. Sloane, N. J. A. (ed.). "Sequence A157017". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.