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| viXra.org > Number Theory > viXra:2203.0170 |
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Number Theory |
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Authors: Theophilus Agama
In this paper we introduce and develop the notion of spanning of integers along functions $f:mathbb{N}longrightarrow mathbb{R}$. We apply this method to a class of problems requiring to determine if the equations of the form $tf(n)=n-k$ has a solution $nin mathbb{N}$ for a fixed $kin mathbb{N}$ and some $tin mathbb{N}$. In particular, we show that begin{align}# {nleq s~|~tvarphi(n)+1=n,~mathbf{for~some}~tin mathbb{N}}geq frac{s}{2log s}prod limits_{p | lfloor sfloor }(1-frac{1}{p})^{-1}-frac{3}{2}e^{gamma}onumberend{align}for $sgeq s_o$, where $varphi$ is the Euler totient function and $gamma=0.5772cdots$ is the Euler-Macheroni constant
Comments: 7 Pages.
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[v1] 2022-03-29 06:52:17
[v2] 2024-06-28 20:59:06
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