Catalan数和Narayana数的完全幂
Perfect powers in Catalan and Narayana numbers
摘要：
当Catalan数或Narayana数是（非平凡的）完全幂时？对于Catalan数，我们证明了答案是“永远不会”。然而，我们证明，对于每个b，Narayana数N（a，b）是一个（非平凡）完满平方的无限多个值的a，我们展示了如何计算所有的。我们还猜想，N（a，b）是从来没有一个（非平凡）完美的k次幂k大于2，我们证明了这个猜想的一些情况。
英文摘要：
When a Catalan number or a Narayana number is a (non-trivial) perfect power? For Catalan numbers, we show that the answer is "never". However, we prove that for every b, the Narayana number N(a,b) is a (non-trivial) perfect square for infinitely many values of a, and we show how to compute all of them. We also conjecture that N(a,b) is never a (non-trivial) perfect k-th power for k greater than 2 and we prove some cases of this conjecture.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
