论文标题
某些理性权力的扰动序列和锡拉丘兹问题的应用的收敛
Convergence of some perturbed sequences of rational powers and application to syracuse problem
论文作者
论文摘要
Sequences of rational powers \left( ξ\left( \frac{p}{q} \right)^{n} \right)_{n\ge 0}, especially in the case \frac{p}{q}=\frac{3}{2}, have a connection with many important combinatorics and number theory problems as for example Syracuse, Z-number and警告问题。已知来自此类问题的猜想是棘手的，迄今为止，只有很少的部分结果存在。在本文中，我们研究了形式\ left（s_ {n} = \ left（ξ+σ_{n} \ right）\ left（\ frac {p^{n}}} {q^^n+e_ {假设此类序列是确定性的并且它们具有控制的阳性扰动，我们确定了收敛结果：min_ {n \ ge 0}（s_ {n}）\ le q^{2}。作为应用程序，我们表明锡拉丘兹序列是“分支序列”，具有收敛的所有必需条件，因此确认了Collatz的猜想。 关键字：理性力量的序列，锡拉丘兹的猜想，Collatz问题，3x+1问题。
Sequences of rational powers \left( ξ\left( \frac{p}{q} \right)^{n} \right)_{n\ge 0}, especially in the case \frac{p}{q}=\frac{3}{2}, have a connection with many important combinatorics and number theory problems as for example Syracuse, Z-number and waring problems. Conjectures from such problems are known to be intractable and only few partial results exist until now. In this paper, we study a family of perturbed sequences of rational powers called 'Branch sequences' of the form \left( S_{n}=\left( ξ+Σ_{n} \right)\left( \frac{p^{n}}{q^{n+e_{n}}} \right) \right)_{n\ge 0}. Under the assumption that such sequences are deterministic and they have controlled positive perturbations, we establish the convergence result: min_{n\ge 0}(S_{n})\le q^{2}. As an application, we show that Syracuse sequences are 'Branch sequences' with all the required conditions for convergence and therefore this confirms the Collatz conjecture. Keywords: Sequences of rational powers, Syracuse conjecture, Collatz problem, 3x+1 problem.
