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where f (2k−1) is the (2k − 1)th derivative of f and B 2k is the (2k)th Bernoulli number: B 2 = 1 / 6 , B 4 = − + 1 / 30 , and so on. Setting f ( x ) = x , the first derivative of f is 1, and every other term vanishes, so [ 15 ]
For instance, the first counterexample must be odd because f(2n) = n, smaller than 2n; and it must be 3 mod 4 because f 2 (4n + 1) = 3n + 1, smaller than 4n + 1. For each starting value a which is not a counterexample to the Collatz conjecture, there is a k for which such an inequality holds, so checking the Collatz conjecture for one starting ...
Mathematics. Four is the smallest composite number, its proper divisors being 1 and 2. [ 4] Four is the sum and product of two with itself: , the only non-zero number such that , which also makes four the smallest and only even squared prime number and hence the first squared prime of the form , where is a prime.
Later X-planes supported important research in a multitude of aerodynamic and technical fields, but only the North American X-15 rocket plane of the early 1960s achieved comparable fame to that of the X-1. [citation needed] X-planes 8, 9, 11, 12, and 17 were actually missiles [8] used to test new types of engines, and some other vehicles were ...
In linguistics, X-bar theory is a model of phrase-structure grammar and a theory of syntactic category formation [1] that was first proposed by Noam Chomsky in 1970 [2] reformulating the ideas of Zellig Harris (1951 [3]), and further developed by Ray Jackendoff (1974, [4] 1977a, [5] 1977b [6]), along the lines of the theory of generative grammar put forth in the 1950s by Chomsky.
If Friday's July jobs report reveals the unemployment rate rose to 4.2% during the month, the Sahm Rule would be triggered. But economists, including Sahm herself, are cautious about such an ...
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