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239 (number)

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Title: 239 (number)  
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Subject: List of prime numbers, 238 (number), 240 (number), Sophie Germain prime, 230 (number)
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239 (number)

238 239 240
Cardinal two hundred thirty-nine
Ordinal 239th
(two hundred and thirty-ninth)
Factorization 239
Prime yes
Roman numeral CCXXXIX
Binary 111011112
Ternary 222123
Quaternary 32334
Quinary 14245
Senary 10356
Octal 3578
Duodecimal 17B12
Hexadecimal EF16
Vigesimal BJ20
Base 36 6N36

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 − 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.[1]

In other fields

239 is also:


  1. ^  A157017
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