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Integer dynamics
International Journal of Number Theory ( IF 0.5 ) Pub Date : 2021-07-17
Dino Lorenzini, Mentzelos Melistas, Arvind Suresh, Makoto Suwama, Haiyang Wang

Let b2 be an integer, and write the base b expansion of any non-negative integer n as n=x0+x1b++xdbd, with xd>0 and 0xi<b for i=0,,d. Let ϕ(x) denote an integer polynomial such that ϕ(n)>0 for all n>0. Consider the map Sϕ,b:00, with Sϕ,b(n):=ϕ(x0)++ϕ(xd). It is known that the orbit set {n,Sϕ,b(n),Sϕ,b(Sϕ,b(n)),,} is finite for all n>0. Each orbit contains a finite cycle, and for a given b, the union of such cycles over all orbit sets is finite.

Fix now an integer 1 and let ϕ(x)=x2. We show that the set of bases b2 which have at least one cycle of length always contains an arithmetic progression and thus has positive lower density. We also show that a 1978 conjecture of Hasse and Prichett on the set of bases with exactly two cycles needs to be modified, raising the possibility that this set might not be finite.



中文翻译:

整数动力学

2 是一个整数,并写下基数 任何非负整数的展开 n 作为 n=X0+X1++Xdd, 和 Xd>00X一世< 为了 一世=0,,d. 让φ(X) 表示一个整数多项式,使得 φ(n)>0 对所有人 n>0. 考虑地图φ,00, 和 φ,(n)=φ(X0)++φ(Xd). 众所周知,轨道集{n,φ,(n),φ,(φ,(n)),,} 对所有人都是有限的 n>0. 每个轨道包含一个有限周期,对于给定的,所有轨道集上的这些循环的并集是有限的。

现在修复一个整数 1 然后让 φ(X)=X2. 我们证明了基集2 至少有一个周期的长度 总是包含一个等差数列,因此具有正的较低密度。我们还表明,Hasse 和 Prichett 在 1978 年关于恰好有两个循环的基集的猜想需要修改,这增加了该集可能不是有限集的可能性。

更新日期:2021-07-19
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