Ternary Egyptian fractions with prime denominator

Adva Mond; Julien Portier

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Ternary Egyptian fractions with prime denominator

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arxiv 2201.11805 v3 pith:IHBA5GVE submitted 2022-01-27 math.NT math.CO

Ternary Egyptian fractions with prime denominator

Adva Mond , Julien Portier This is my paper

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keywords fracmathbbprimebounddenominatoregyptianexistsfractions

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For a prime number $p$, let $A_3(p)= | \{ m \in \mathbb{N}: \exists m_1,m_2,m_3 \in \mathbb{N}, \frac{m}{p}=\frac{1}{m_1}+\frac{1}{m_2}+\frac{1}{m_3} \} |$. In 2019 Luca and Pappalardi proved that $x (\log x)^3 \ll \sum_{p \le x} A_{3}(p) \ll x (\log x)^5$. We improve the upper bound, showing $\sum_{p \le x} A_{3}(p) \ll x (\log x)^3 (\log \log x)^2$.

Ternary Egyptian fractions with prime denominator

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