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My question is: Do we have an splitting equation where we can produce fractions with odd denominators?

To split an Egyptian fraction to Egyptian fractions, we can use the splitting equation below:

$\frac{1}{n}= \frac{1}{n+1}+\frac{1}{n(n+1)}$

The key limitation of the above equation is the following:

If $n$ is even, then $n+1$ is odd and $n(n+1)$ is even, otherwise $n+1$ is even and $n(n+1)$ is even.

Either way, the splitting equation produces with at least one even Egyptian fraction.

An example of a splitting to Odd Egyptian fraction is given below:

$\frac{1}{3}= \frac{1}{5}+\frac{1}{9}+\frac{1}{45}$

$\frac{1}{5}= \frac{1}{9}+\frac{1}{15}+\frac{1}{45}$

$\frac{1}{7}= \frac{1}{15}+\frac{1}{21}+\frac{1}{35}$

$\frac{1}{7}= \frac{1}{9}+\frac{1}{45}+\frac{1}{105}$

$\frac{1}{9}= \frac{1}{15}+\frac{1}{35}+\frac{1}{63}$

$\frac{1}{11}= \frac{1}{21}+\frac{1}{33}+\frac{1}{77}$

The link below is useful for further details about egyptian fraction: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html#section9.5

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A general solution is for every positive positive integer $\ n\ $ :

  • If $n$ is odd , then $$\frac{1}{3n+2}+\frac{1}{6n+3}+\frac{1}{18n^2+21n+6}=\frac{1}{2n+1}$$ is a solution with odd denominators

  • If $n$ is even , then $$\frac{1}{3n+3}+\frac{1}{6n+3}+\frac{1}{6n^2+9n+3}=\frac{1}{2n+1}$$ is a solution with odd denominators

So, for every odd $\ k\ge 3\ $ we can write $\ \frac 1k\ $ with $\ 3\ $ distinct fractions with odd denominators.

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  • $\begingroup$ thank you for the answer. Can we not have only one equation for both odd and even $n$? Or it is impossible to construct one? $\endgroup$ Mar 18, 2020 at 17:54
  • $\begingroup$ Maybe, but I have no idea how. $\endgroup$
    – Peter
    Mar 18, 2020 at 20:43
  • $\begingroup$ @Peter: Do you happen to have a reference for your two equations? $\endgroup$ Mar 19, 2020 at 7:23
  • $\begingroup$ @JoseArnaldoBebita-Dris No, I found them by analyzing concrete solutions. $\endgroup$
    – Peter
    Mar 19, 2020 at 8:28
  • $\begingroup$ Okay, thanks! @Peter $\endgroup$ Mar 19, 2020 at 8:35
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Just a thought:

If you limit the problem to just the examples you gave; that is, can any given odd Egyptian fraction be split into three different odd Egyptian fractions?

The examples you gave have two main algebraic forms:

$$\frac{1}{n}=\frac{1}{b}+\frac{1}{an}+\frac{1}{abn}\tag{1}$$

and

$$\frac{1}{n}=\frac{1}{ab}+\frac{1}{an}+\frac{1}{bn}\tag{2}$$

where $a$, $b$ and $n$ are all odd positive integers.

Then one way of proceeding is to study congruence patterns for the two basic forms (1) and (2), for example for $n=4m+1$ and $n=4m+3$ in the first instance.

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