Some facts about number 239

Some facts about number 239

I've got this info from Marina Krol (1966, 5) ( marina_krol@smtplink.mssm.edu).
I've found this interesting. :-)

Work reported herein was conducted at the Artificial Intelligence
Laboratory, a Massachusetts Institute of Technology research program
supported in part by the Advanced Research Projects Agency of the
Department of Defense and monitored by the Office of Naval Research
under Contract Number N00014-70-A-0362-0002.

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ITEM 63 (Schroeppel, etc.):

The joys of 239 are as follows: 

* pi = 16 arctan (1/5) - 4 arctan(1/239), 
* which is related to the fact that 2 * 13^4 - 1 = 239^2, 
* which is why 239/169 is an approximant (the 7th) of sqrt(2). 
* arctan(1/239) = arctan(1/70) - arctan(1/99) = arctan(1/408) + 
arctan(1/577) 
* 239 needs 4 squares (the maximum) to express it. 
* 239 needs 9 cubes (the maximum, shared only with 23) to express it. 
* 239 needs 19 fourth powers (the maximum) to express it. 
* (Although 239 doesn't need the maximum number of fifth powers.) 
* 1/239 = .00418410041841..., which is related to the fact that 
* 1,111,111 = 239 * 4,649. 
* 239 = 11101111 base 2. 
* 239 = 22212 base 3. 
* 239 = 3233 base 4. 
* There are 239 primes < 1500. 
* K239 is Mozart's only work for 2 orchestras. 
* Guess what memo this is. 
* And 239 is prime, of course.

Added by Alex Glushkovsky:
239 = 16^2-16^1-16^0
where 16 = (2-1)(3-1)(9-1)
also:
239 = 2^8-2^4-2^0 = 2^(2x2x2)-2^(2x2)-2^(2-2)
 

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