Collatz conjecture - Statement of the problem

Consider the following operation on an arbitrary positive integer: If the number is even, divide it by two. If the number is odd, triple it and add one. For example, if this operation is performed on 3, the result is 10; if it is performed on 28, the result is 14. In mathematical notation, define the function f as follows: Now, form a sequence by performing this operation repeatedly, beginning with any positive integer, and taking the result at each step as ...

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Collatz conjecture, Collatz conjecture - Statement of the problem, Collatz conjecture - Examples, Collatz conjecture - Program to calculate Collatz sequences, Collatz conjecture - Supporting arguments, Collatz conjecture - Experimental evidence, Collatz conjecture - Probabilistic evidence, Collatz conjecture - Other ways of looking at it, Collatz conjecture - In reverse, Collatz conjecture - As rational numbers, Collatz conjecture - As an abstract machine, Collatz conjecture - As iterating a real or complex map, Collatz conjecture - Optimizations

Read more here: » Collatz conjecture: Encyclopedia II - Collatz conjecture - Statement of the problem

The Collatz conjecture is an unresolved conjecture in mathematics. It is named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanislaw Ulam), the Syracuse problem, as the hailstone sequence or hailstone numbers, or as Wondrous numbers as per Gödel, Escher, Bach. It asks whether a certain kind of number sequence always ...

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• Collatz conjecture - Statement of the problem
• Collatz conjecture - Examples
• Collatz conjecture - Program to calculate Collatz sequences
• Collatz conjecture - Supporting arguments
• Collatz conjecture - Experimental evidence
• Collatz conjecture - Probabilistic evidence
• Collatz conjecture - Other ways of looking at it
• Collatz conjecture - In reverse
• Collatz conjecture - As rational numbers
• Collatz conjecture - As an abstract machine
• Collatz conjecture - As iterating a real or complex map
• Collatz conjecture - Optimizations

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Collatz conjecture - In reverse. There is another approach to prove the following conjecture, which considers the bottom-up method of growing the Collatz graph. The Collatz graph is defined by an inverse relation, So, instead of proving that all natural numbers eventually lead to 1, we can prove that 1 leads to all natural numbers. Also, the inverse relation forms a tree except for the 1-2 loop. Note that the relation being inverted here is (3n + 1) / 2 (see Optimizations below). ...

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Collatz conjecture, Collatz conjecture - Statement of the problem, Collatz conjecture - Examples, Collatz conjecture - Program to calculate Collatz sequences, Collatz conjecture - Supporting arguments, Collatz conjecture - Experimental evidence, Collatz conjecture - Probabilistic evidence, Collatz conjecture - Other ways of looking at it, Collatz conjecture - In reverse, Collatz conjecture - As rational numbers, Collatz conjecture - As an abstract machine, Collatz conjecture - As iterating a real or complex map, Collatz conjecture - Optimizations

Read more here: » Collatz conjecture: Encyclopedia II - Collatz conjecture - Other ways of looking at it

If n is a multiple of 4, it can be divided by 4. Reason: It is initially even. When it is divided by two, it is still even. More generally, one can write n as the product of its factors, and change the power of 2 to 0. Reason: If the power of 2 of the prime factorization is greater than 0, the number is even, and the following step would produce the same factorization with one less power of 2. For instance: instead of 15, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, ...

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Collatz conjecture, Collatz conjecture - Statement of the problem, Collatz conjecture - Examples, Collatz conjecture - Program to calculate Collatz sequences, Collatz conjecture - Supporting arguments, Collatz conjecture - Experimental evidence, Collatz conjecture - Probabilistic evidence, Collatz conjecture - Other ways of looking at it, Collatz conjecture - In reverse, Collatz conjecture - As rational numbers, Collatz conjecture - As an abstract machine, Collatz conjecture - As iterating a real or complex map, Collatz conjecture - Optimizations

Read more here: » Collatz conjecture: Encyclopedia II - Collatz conjecture - Optimizations

Although the conjecture has not been proven, most mathematicians who have looked into the problem believe intuitively that the conjecture is true. Here are some reasons for expecting this. Collatz conjecture - Experimental evidence. The conjecture has been checked by computer for all start values up to 3 × 253 = 27,021,597,764,222,976 [1]. Intuitively, it would be surprising if the smallest counterexample were larger than this large number. As computers speed up and testing techniques improve, this testing horizon will continue to increase. See also:

Collatz conjecture, Collatz conjecture - Statement of the problem, Collatz conjecture - Examples, Collatz conjecture - Program to calculate Collatz sequences, Collatz conjecture - Supporting arguments, Collatz conjecture - Experimental evidence, Collatz conjecture - Probabilistic evidence, Collatz conjecture - Other ways of looking at it, Collatz conjecture - In reverse, Collatz conjecture - As rational numbers, Collatz conjecture - As an abstract machine, Collatz conjecture - As iterating a real or complex map, Collatz conjecture - Optimizations

Read more here: » Collatz conjecture: Encyclopedia II - Collatz conjecture - Supporting arguments

A specific Collatz sequence can be easily computed, as is shown by this pseudocode example: def collatz(n) show n if n.odd? and n > 1 collatz(3n + 1) else if n.even? collatz(n / 2) This program halts when the sequence reaches 1, in order to avoid printing an endless cycle of 4, 2, 1. If the Collatz conjecture is true, the program will always halt no matter what positive starting integer is given to it. (See Halting problem for a discussion of the relation ...

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Collatz conjecture, Collatz conjecture - Statement of the problem, Collatz conjecture - Examples, Collatz conjecture - Program to calculate Collatz sequences, Collatz conjecture - Supporting arguments, Collatz conjecture - Experimental evidence, Collatz conjecture - Probabilistic evidence, Collatz conjecture - Other ways of looking at it, Collatz conjecture - In reverse, Collatz conjecture - As rational numbers, Collatz conjecture - As an abstract machine, Collatz conjecture - As iterating a real or complex map, Collatz conjecture - Optimizations

Read more here: » Collatz conjecture: Encyclopedia II - Collatz conjecture - Program to calculate Collatz sequences

For instance, starting with n = 6, one gets the sequence 6, 3, 10, 5, 16, 8, 4, 2, 1. (The Collatz conjecture says that this process always reaches 1, no matter what the starting value.) Starting with n = 11, the sequence takes longer to reach 1: 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. If the starting value n = 27 is chosen, the sequence takes 112 step ...

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Collatz conjecture, Collatz conjecture - Statement of the problem, Collatz conjecture - Examples, Collatz conjecture - Program to calculate Collatz sequences, Collatz conjecture - Supporting arguments, Collatz conjecture - Experimental evidence, Collatz conjecture - Probabilistic evidence, Collatz conjecture - Other ways of looking at it, Collatz conjecture - In reverse, Collatz conjecture - As rational numbers, Collatz conjecture - As an abstract machine, Collatz conjecture - As iterating a real or complex map, Collatz conjecture - Optimizations

Read more here: » Collatz conjecture: Encyclopedia II - Collatz conjecture - Examples