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239 (number) - Wikipedia
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Natural number
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. [1] 239 is a Sophie Germain prime and a Newman–Shanks–Williams prime.[2] It is an Eisenstein prime with no imaginary part and real part of the form 3n − 1 (with no exponentiation implied). 239 is a factor of the repdigit 1111111, with the other prime factor being 4649. 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.[3]
HAKMEM (incidentally AI memo 239 of the MIT AI Lab) included an item on the properties of 239, including these:[4]
- 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.[5]
- 239/169 is a convergent of the simple 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.[6]
- The only solutions of the Diophantine equation y2 + 1 = 2x4 in positive integers are (x, y) = (1, 1) or (13, 239).
- ^ Sloane, N. J. A. (ed.). "Sequence A109611 (chen prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ 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.
- ^ Baker, Henry (April 1995). "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". inwap.com. Retrieved 2025-01-29.
- ^ 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.
Integers −1 0s
100s
200s
300s
400s
500s
600s
700s
800s
900s
1000s
- 100,000
- 1,000,000
- 10,000,000
- 100,000,000
- 1,000,000,000
- 10,000,000,000
- 100,000,000,000
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