共 1 条
[1]
Shaw DJ, 2006, FIBONACCI QUART, V44, P194
A proof of the Collatz conjecture
被引：6
作者：
Bruckman, Paul S.
[1
]
机构：
[1] POB 150, Sointula, BC V0N 3E0, Canada
关键词：
U;
the modified Collatz transformation;
T-k;
T- the kth convolution of the Collatz transformation;
{n(0);
U(n(0));
U-2(n(0));
.;
any Collatz chain;
N-k = U-k(N-0);
where N-0 is the smallest assumed starting value of an infinite Collatz chain;
{E-k} the sequence of exponents associated with N-0;
{M-k};
the maximal sequence of exponents associated with N-0;
S-k = 2(Ek)N(k) - 3(k)N(0) = Sigma(k-1)(j=0) 3(k-1-j)2(Ej);
D O I：
10.1080/00207390701691574
中图分类号：
G40 [教育学];
学科分类号：
040101 ;
120403 ;
摘要：
An elementary proof by contradiction of the Collatz Conjecture (CC) (also known as the '3X+1' Conjecture), is presented. A modified form of the Collatz transformation is formulated, leading to the concept of a modified Collatz chain. A smallest counterexample N-0 is hypothesized; the existence of N-0 implies that N-0 must generate an infinite sequence {N-k}, each of whose elements is at least as large as N-0. A formula for Nk is derived, in terms of an auxiliary sequence {E-k} and the starting value N-0. It is shown that each Ek satisfies k <= E-k < 1.585k; this, in turn, leads us to conclude that N-0 is unbounded, which is a contradiction of its definition, thereby establishing CC.
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页码：403 / 407
页数：7
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