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# On the Collatz $3n + 1$ Algorithm

Lynn E. Garner
Proceedings of the American Mathematical Society
Vol. 82, No. 1 (May, 1981), pp. 19-22
DOI: 10.2307/2044308
Stable URL: http://www.jstor.org/stable/2044308
Page Count: 4
The number theoretic function $s(n) = \frac{1}{2} n$ if $n$ is even, $s(n) = 3n + 1$ if $n$ is odd, generates for each $n$ a Collatz sequence $\{ s^k(n)\}^\infty_{k = 0}, s^0(n) = n, s^k(n) = s(s^{k - 1}(n))$. It is shown that if a Collatz sequence enters a cycle other than the $4, 2, 1, 4,\ldots$ cycle, then the cycle must have many thousands of terms.