The 3x + 1 Conjugacy Map
The 3x + 1 Conjugacy Map
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3x 1 共轭图
DOI：
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发表时间：
1996
期刊：
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通讯作者：
Jeffrey C. Lagarias
中科院分区：
文献类型：
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作者：
D. Bernstein;Jeffrey C. Lagarias
对于3x+1映射T和移位映射S定义为：对于x为奇数T(x) = (3x +1)/2，对于x为偶数T(x) = x/2，对于x为奇数S(x) = (x−1)/2，对于x为偶数S(x) = x/2。2进整数z2上的3x + 1共轭映射Φ将S共轭到T上，即Φ o S o Φ-1 = T。映射Φ mod 2n在Z/2 n Z上诱导出一个排列Φ n。特别地，我们证明了当n≥6时它的阶为2n−4。我们也计算Φ n的1个循环对于n到1000；结果表明Φ恰好有两个奇数不动点。结果推广到ax + b映射，其中ab是奇数。
Abstract The 3x+1 map T and the shift map S are defined by T(x) = (3x + 1)/2 for x odd, T(x) = x/2 for x even, while S(x) = (x − 1)/2 for x odd, S(x) = x/2 for x even. The 3x + 1 conjugacy map Φ on the 2-adic integers Z 2 conjugates S to T, i.e., Φ o S o Φ-1 = T. The map Φ mod 2n induces a permutation Φ n on Z/2 n Z. We study the cycle structure of Φ n . In particular we show that it has order 2 n − 4 for n ≥ 6. We also count 1-cycles of Φ n for n up to 1000; the results suggest that Φ has exactly two odd fixed points. The results generalize to the ax + b map, where ab is odd.