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From Wikipedia, the free encyclopedia

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; 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 − 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.[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:

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.
This page was last edited on 26 March 2019, at 14:43
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