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While virtually all real numbers k will eventually have infinitely many convergents m / n whose distance from k is significantly smaller than this limit, the convergents for φ (i.e., the numbers 5 / 3 , 8 / 5 , 13 / 8 , 21 / 13 , etc.) consistently "toe the boundary", keeping a distance of almost exactly ...
For example, Turing's 1936 paper (p. 263) observed that since the halting problem for each Turing machine is equivalent to a first-order logical formula of form , the problem () is undecidable. The precise boundaries are known, sharply:
The first 3 powers of 2 with all but last digit odd is 2^4 = 16, 2^5 = 32 and 2^9 = 512. The next such power of 2 of form 2^n should have n of at least 6 digits. The only powers of 2 with all digits distinct are 2^0 = 1 to 2^15 = 32768, 2^20 = 1048576 and 2^29 = 536870912.
Gottfried Wilhelm Leibniz (or Leibnitz; [a] 1 July 1646 [O.S. 21 June] – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to many other branches of mathematics, such as binary arithmetic, and statistics.
As a consequence, for every integer d > 1 there are either 4 or 5 Fibonacci numbers with d decimal digits. More generally, in the base b representation, the number of digits in F n is asymptotic to n log b φ = n log φ log b . {\displaystyle n\log _{b}\varphi ={\frac {n\log \varphi }{\log b}}.}
In 2002 production rose to nearly 3.25 million and then jumped to 4.44 million in 2003, 5.07 million in 2004, 5.71 million in 2005, 7.28 million in 2006, 8.88 million in 2007, 9.35 million in 2008 and 13.83 million in 2009. China has become the number-one automaker in the world as of 2009. Domestic sales have kept pace with production.
Of those identifying as Māori at the 2018 census, 352,755 people (45.5%) identified as of sole Māori ethnicity while 336,174 people (43.3%) identified as of both European and Māori ethnicity, due to the high rate of intermarriage between the two ethnicities.
Isaac Asimov (/ ˈ æ z ɪ m ɒ v / AZ-ih-mov; [b] c. January 2, 1920 [a] – April 6, 1992) was an American writer and professor of biochemistry at Boston University.During his lifetime, Asimov was considered one of the "Big Three" science fiction writers, along with Robert A. Heinlein and Arthur C. Clarke. [2]