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If a term is odd, the next term is 3 times the previous term plus 1. The conjecture is that these sequences always reach 1, no matter which positive integer is chosen to start the sequence. The conjecture has been shown to hold for all positive integers up to 2.95 × 10 20, but no general proof has been found.
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A counter-based version of Middle-Square Weyl Sequence RNG. Similar to Philox in design but significantly faster. Collatz-Weyl Generators 2023 Tomasz R. Działa [40] A class of chaotic counter-based generators applying a broad class of non-invertible generalized Collatz mappings and Weyl sequences, enabling the generation of multiple ...
The following other wikis use this file: Usage on ca.wikipedia.org Conjectura de Collatz; Usage on eu.wikipedia.org Collatzen aierua; Usage on fi.wikipedia.org
The sequence of gaps between consecutive prime numbers has a finite lim inf. See Polymath Project#Polymath8 for quantitative results. 2013: Adam Marcus, Daniel Spielman and Nikhil Srivastava: Kadison–Singer problem: functional analysis: The original problem posed by Kadison and Singer was not a conjecture: its authors believed it false.
Directed graph showing the orbits of the first 1000 numbers in the Collatz conjecture. The integers from 1 to 1000 are colored from red to violet according to their value. French
1, 2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, ... For n ≥ 2, a ( n ) is the prime that is finally reached when you start with n , concatenate its prime factors (A037276) and repeat until a prime is reached; a ( n ) = −1 if no prime is ever reached.
In contrast, look at how useful the heuristic is: every odd number in a Collatz sequence will be succeeded by a mean of two even numbers. Earlier, I showed how to predict the number of odd numbers in the Collatz sequence of 2**177149-1 and got an answer (853681) that was within 0.12% of the actual result of 854697.