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On Egyptian fractions of length 3
Volume 62, no. 1
(2021),
pp. 257–274
https://doi.org/10.33044/revuma.1798
Abstract
Let $a,n$ be positive integers that are relatively prime.
We say that $a/n$ can be represented as an Egyptian fraction of length $k$ if
there exist positive integers $m_1, \ldots, m_k$
such that $\frac{a}{n} = \frac{1}{m_1}+ \cdots + \frac{1}{m_k}$.
Let $A_k(n)$ be the number of solutions $a$ to this equation.
In this article, we give a formula for $A_2(p)$ and
a parametrization for Egyptian fractions of length $3$,
which allows us to give bounds to $A_3(n)$,
to $f_a(n) = \#\{(m_1,m_2,m_3) : \frac{a}{n} =
\frac{1}{m_1}+\frac{1}{m_2}+\frac{1}{m_3}\}$, and finally to $F(n) =
\#\{(a,m_1,m_2,m_3) : \frac{a}{n}=\frac{1}{m_1}+\frac{1}{m_2}+\frac{1}{m_3}\}$.
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Published by the Unión Matemática Argentina |
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