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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 ...
Secretary problem. Graphs of probabilities of getting the best candidate (red circles) from n applications, and k / n (blue crosses) where k is the sample size. The secretary problem demonstrates a scenario involving optimal stopping theory [1] [2] that is studied extensively in the fields of applied probability, statistics, and decision theory ...
Lucas numbers have L 1 = 1, L 2 = 3, and L n = L n−1 + L n−2. Primefree sequences use the Fibonacci recursion with other starting points to generate sequences in which all numbers are composite. Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The Pell numbers have P n = 2P n−1 + P n−2.
Take X to be a partially ordered set. If T 1, T 2 are rooted trees with vertices labeled in X, we say that T 1 is inf-embeddable in T 2 and write if there is an injective map F from the vertices of T 1 to the vertices of T 2 such that: For all vertices v of T 1, the label of v precedes the label of ();
If F 1 were in fact inconsistent, then F 2 would prove for some n that n is the code of a contradiction in F 1. But if F 2 also proved that F 1 is consistent (that is, that there is no such n), then it would itself be inconsistent. This reasoning can be formalized in F 1 to show that if F 2 is consistent, then F 1 is consistent.
Knuth's up-arrow notation. In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976. [1] In his 1947 paper, [2] R. L. Goodstein introduced the specific sequence of operations that are now called hyperoperations. Goodstein also suggested the Greek names tetration, pentation ...
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