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

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. After all, the key is that infinite numerical iteration is quite difficult.

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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. Numbers with a total stopping time longer than that of any smaller starting value form a sequence beginning with:.

The only known cycle is 1,2 of period 2, called the trivial cycle, 3x-1.

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Solve Practice Play. Game Central. Greatest Common Factor. Least Common Multiple. Order of Operations. Mixed Fractions.

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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 Let be an integer.

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Using this form for f n , it can be shown that the parity sequences for two numbers m and n will agree in the first k terms if and only if m and n are equivalent modulo 2 k. Caraiani, A. In , John Horton Conway proved that a natural generalization of the Collatz problem is algorithmically undecidable. The only known cycle is 1,2 of period 2, called the trivial cycle. Krasikov, I. Hidden category: Articles containing conjectures. An extension to the Collatz conjecture is to include all integers, not just positive integers. The following mathematical induction proves that row number of 4 r is. Since it was put forward, the conjecture has never been stopped studying on it. Lemma 5. A closely related fact is that the Collatz map extends to the ring of 2-adic integers , which contains the ring of rationals with odd denominators as a subring. Science and Complexity, 25, Therefore, lemma 5. JSTOR The Collatz conjecture [a] is one of the most famous unsolved problems in mathematics.

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.

Monks, K. These cycles are listed here, starting with the well-known cycle for positive n :. The Collatz conjecture equivalently states that this tag system, with an arbitrary finite string of a as the initial word, eventually halts see Tag system for a worked example. The following table gives the sequences resulting from iterating the Collatz function starting with the first few pure hailstone numbers see A Hailstone sequences can be computed by the 2-tag system with production rules. Discrete Impuls Systems. An extension to the Collatz conjecture is to include all integers, not just positive integers. Joseph Pe, L. In this system, the positive integer n is represented by a string of n copies of a , and iteration of the tag operation halts on any word of length less than 2. Advances in Applied Mathematics, 45, With this, they show that no integer is in a Baker domain , which implies that any integer is either eventually periodic or belongs to a wandering domain. For even numbers, divide by 2; For odd numbers, multiply by 3 and add 1.