A new statistic for the \(3x+1\) problem (Q2781286)
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scientific article; zbMATH DE number 1721031
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A new statistic for the \(3x+1\) problem |
scientific article; zbMATH DE number 1721031 |
Statements
19 March 2002
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\(3n+1\) problem
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\(3n+1\) conjecture
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Collatz problem
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random behavior of \(3n+1\) trajectories
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A new statistic for the \(3x+1\) problem (English)
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The paper contains a proof of the following assertion: Let \(a_1,\ldots,a_n\) be a \(3x+1\) trajectory, i.e.\ \(a_{j+1}=a_j/2\) if \(a_j\) is even and \(=3a_j+1\) if \(a_j\) is odd, with \(a_1=m\) and \(a_n=1\), and assume \(m\geq 3\). Then the quotient NEWLINE\[NEWLINE C(m)={a_1a_2+\ldots+a_{n-1}a_n+a_na_1\over a_1^2+\ldots+a_n^2} NEWLINE\]NEWLINE satisfies the sharp estimates \(9/13<C(m)<5/7\). Moreover, experimental results concerning the distribution of \(C(m)\) are given that lead to interesting open problems.
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