On A Splitting Equation of an Egyptian fraction to Egyptian fractions such that all produced fractions have odd denominators.

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

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.

• 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? Mar 18, 2020 at 17:54
• Maybe, but I have no idea how. Mar 18, 2020 at 20:43
• @Peter: Do you happen to have a reference for your two equations? Mar 19, 2020 at 7:23
• @JoseArnaldoBebita-Dris No, I found them by analyzing concrete solutions. Mar 19, 2020 at 8:28
• Okay, thanks! @Peter Mar 19, 2020 at 8:35

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.