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239 (two hundred () 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 3''n'' − 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 ''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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「239 (number)」の詳細全文を読む スポンサード リンク
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