Abstract:Büyükaşik et al. [Publ. Math. Debrecen, 2024] introduced a family of generalizations of Euler's totient function $\varphi(n)$, by setting $\varphi_k(n) = \sum_{a} a^k$ for $a \in [1, n]$ such that $(a, n) = 1$, with $\varphi_0(n) = \varphi(n)$. Letting $\mathcal{D}_{s} = \{ k \geq s : \forall n \geq 1 \ \varphi_s(n) \mid \varphi_k(n) \}$, Büyükaşik et al. proved that $\mathcal{D}_{s}$ is finite for each $s \geq 0$, and conjectured that $\mathcal{D}_{1} = \{ 1, 3, 15 \}$ and provided computations to support this conjecture. We succeed in proving this conjecture, using an argument based on our extensive interactions with GPT-5.5 Pro.
From: John Campbell [view email]
[v1]
Mon, 1 Jun 2026 03:34:54 UTC (8 KB)
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