Zeckendorf定理的弱逆
The weak converse of Zeckendorf's Theorem
摘要：
根据Zeckendorf定理，每一个正整数都可以被唯一地写成斐波那契数列的不相邻项之和，其逆命题是，如果正整数中的一个数列具有这个性质，它一定是斐波那契数列。如果我们转而考虑寻找具有这种性质的单调序列的问题，我们称之为Zeckendorf定理的弱逆。本文首先介绍了Zeckendorf条件的一个推广，然后介绍了Zeckendorf定理及其在一般Zeckendorf条件下的弱逆。我们还将推广和结果推广到区间$(0,1)$中的实数和$p$ -adic整数。
英文摘要：
By Zeckendorf's Theorem, every positive integer is uniquely written as a sum of non-adjacent terms of the Fibonacci sequence, and its converse states that if a sequence in the positive integers has this property, it must be the Fibonacci sequence. If we instead consider the problem of finding a monotone sequence with such a property, we call it the weak converse of Zeckendorf's theorem. In this paper, we first introduce a generalization of Zeckendorf conditions, and subsequently, Zeckendorf's theorems and their weak converses for the general Zeckendorf conditions. We also extend the generalization and results to the real numbers in the interval $(0,1)$, and to $p$-adic integers.
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