3x-1

It is also known as 3x-1 Collatz problem or the hailstone problem.

It probably came into being between the s and s. In his review paper, J. The problem is traditionally credited to Lothar Collatz, at the University of Hamburg. Since it was put forward, the conjecture has never been stopped studying on it. Up to now, many papers on this conjecture have been published at home and abroad [2] - [11], we can see from these papers [2] [3] [4] [5] that many people limited and stayed on the idea of function iteration.

3x-1

The Collatz conjecture [a] is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. It concerns sequences of integers in which each term is obtained from the previous term as follows: if the previous term is even , the next term is one half of the previous term. If the previous 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. It is named after the mathematician Lothar Collatz , who introduced the idea in , two years after receiving his doctorate. Consider the following operation on an arbitrary positive integer :. Now form a sequence by performing this operation repeatedly, beginning with any positive integer, and taking the result at each step as the input at the next. The Collatz conjecture is: This process will eventually reach the number 1, regardless of which positive integer is chosen initially. If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence that does not contain 1. Such a sequence would either enter a repeating cycle that excludes 1, or increase without bound. No such sequence has been found. The Collatz conjecture asserts that the total stopping time of every n is finite.

The 3x-1 values having the smallest total stopping time with respect to their number of digits in base 2 are the powers of two since 2 n is halved n times to reach 1, and is never increased. The whole process contains three parts, 3x-1.

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3x-1

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Retrieved 14 March The result of jumping ahead k is given by. Kenneth Monks, M. But of course in the search for a counterexample to the Collatz conjecture, they would have to be programmed to keep track of previous numbers encountered in the sequence to compare them against new values. In , Terence Tao improved this result by showing, using logarithmic density , that almost all in the sense of logarithmic density Collatz orbits are descending below any given function of the starting point, provided that this function diverges to infinity, no matter how slowly. The first thick line towards the middle of the plot corresponds to the tip at 27, which reaches a maximum at Contents move to sidebar hide. Discrete Impuls Systems. But there are also starting values that don't jump up significantly until much later on, closer to the final fall. In the tree graph above, halving steps are denoted by black lines, while blue lines signify tripling steps plus the addition of 1. Proceedings of the 7th Manitoba Conference on Numerical Mathematics. Bibcode : MaCom.. MR Lemma 4. Krasikov, I.

A problem posed by L. Collatz in , also called the mapping, problem, Hasse's algorithm, Kakutani's problem, Syracuse algorithm, Syracuse problem, Thwaites conjecture, and Ulam's problem Lagarias

All Rights Reserved. Unsolved Problems in Number Theory 3rd ed. The corresponding Julia set, shown on the right, consists of uncountably many curves, called hairs , or rays. The starting number 7 is written in base two as Since the powers of two give an element of predictability, it is natural that a tree graph of Collatz sequences put the powers of two on the central axis, or at least line them up. They conjectured that the latter is not the case, which would make all integer orbits finite. Now form a sequence by performing this operation repeatedly, beginning with any positive integer, and taking the result at each step as the input at the next. This leads to the sequence 3, 10, 5, 16, 4, 2, 1, 4, 2, 1, Venturini, G. A is. Note: "Delay records" are total stopping time records.