Let [Formula: see text] be a fixed integer. Define [Formula: see text] to be the unique integer in the range [Formula: see text] which is congruent to [Formula: see text] modulo [Formula: see text]. Given [Formula: see text], let [Formula: see text] and define [Formula: see text] to be the index of the sequence [Formula: see text]. If [Formula: see text] have [Formula: see text] but [Formula: see text] for all proper, non-empty subsets [Formula: see text], then a still open conjecture asserts that [Formula: see text] provided that [Formula: see text]. We give an alternative proof, that does not rely on computer calculations, verifying this conjecture when [Formula: see text] is a product of two prime powers.

中文翻译：

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小长度序列的索引

令 [公式：见正文] 为固定整数。将 [公式：参见文本] 定义为 [公式：参见文本] 范围内的唯一整数，它与 [公式：参见文本] 模 [公式：参见文本] 一致。给定 [Formula: see text]，让 [Formula: see text] 并将 [Formula: see text] 定义为序列 [Formula: see text] 的索引。如果 [Formula: see text] 具有 [Formula: see text] 但 [Formula: see text] 对于所有适当的非空子集 [Formula: see text]，则仍然开放的猜想断言 [Formula: see text] 提供[公式：见正文]。我们给出了一个不依赖于计算机计算的替代证明，当 [公式：见正文] 是两个素数幂的乘积时验证这个猜想。