论文标题
在有关完美权力的一些开放问题上
On some open problems concerning perfect powers
论文作者
论文摘要
我们论文的起点是Kashihara的开放问题编号$ 30 $，涉及OEI的序列$ a001292 $，询问整数的功能有多少个术语。我们确认他的最后一个猜想达到了$ 100128美元的期限，并提供了一般定理，该定理排除了$ 4/9 $的候选人。此外，我们制定了一个涉及OEIS序列$ A352991 $的新的，挑衅的猜想（其中包括$ A001292 $）。我们有风险的猜想指出，属于序列$ a352991 $的所有完美力量都是完美的正方形，如果给定的$ a352991 $不等于一个，则不能将其写成更高秩序的完美力量。该挑战性的猜想已经检查了任何小于$ 101111112131415161718192022222223456789 $的整数的任何整数，到目前为止尚未发现反例。
The starting point of our paper is Kashihara's open problem number $30$, concerning the sequence $A001292$ of the OEIS, asking how many terms are powers of integers. We confirm his last conjecture up to the $100128$-th term and provide a general theorem that rules out $4/9$ of the candidates. Moreover, we formulate a new, provocative, conjecture involving the OEIS sequence $A352991$ (which includes all the terms of $A001292$). Our risky conjecture states that all the perfect powers belonging to the sequence $A352991$ are perfect squares and they cannot be written as higher order perfect powers if the given term of $A352991$ is not equal to one. This challenging conjecture has been checked for any integer smaller than $10111121314151617181920212223456789$ and no counterexample has been found so far.
