I am reading Stillwell's Numbers and Geometry. There is an exercise aboue egyptian fractions which is the following:

I've tried to do it in the following way - Expressing an arbitrary fraction $\frac{n}{m}$ as the sum of two aribtrary egipcian fractions:

$$\frac{1}{a}+\frac{1}{b}=\frac{n}{m}$$

$$\frac{1}{a}+\frac{1}{b}-\frac{n}{m}=0$$

$$\frac{bm+am-abn}{abm}=0$$

Then solving for $a$ yields:

$$a=\frac{b m}{b n-m}$$

And then expressing it as a function:

$$f(b)= \frac{b m}{b n-m}$$

I have tested with $\frac{n}{m}=\frac{3}{4}$ on Mathematica and all the sums of unit fractions I've obtained equals $\frac{3}{4}$. Is this correct? I'm afraid there's something wrong and I'm not seeing it.

I am reading Stillwell's Numbers and Geometry. There is an exercise about Egyptian fractions which is the following:

I've tried to do it in the following way - Expressing an arbitrary fraction $\frac{n}{m}$ as the sum of two aribtrary egipcian fractions:

$$\frac{1}{a}+\frac{1}{b}=\frac{n}{m}$$

$$\frac{1}{a}+\frac{1}{b}-\frac{n}{m}=0$$

$$\frac{bm+am-abn}{abm}=0$$

Then solving for $a$ yields:

$$a=\frac{b m}{b n-m}$$

And then expressing it as a function:

$$f(b)= \frac{b m}{b n-m}$$

I have tested with $\frac{n}{m}=\frac{3}{4}$ on Mathematica and all the sums of unit fractions I've obtained equals $\frac{3}{4}$. Is this correct? I'm afraid there's something wrong and I'm not seeing it.

I am reading Stillwell's Numbers and Geometry. There is an exercise aboue egyptian fractions which is the following:

I've tried to do it in the following way - Expressing an arbitrary fraction $\frac{n}{m}$ as the sum of two aribtrary egipcian fractions:

$$\frac{1}{a}+\frac{1}{b}=\frac{n}{m}$$

$$\frac{1}{a}+\frac{1}{b}-\frac{n}{m}=0$$

$$\frac{bm+am-abn}{abm}=0$$

Then solving for $a$ yields:

$$a=\frac{b m}{b n-m}$$

And then expressing it as a function:

$$f(b)= \frac{b m}{b n-m}$$

I have tested with $\frac{n}{m}=\frac{3}{4}$ on Mathematica and all the sums of unit fractions I've obtained equals $\frac{3}{4}$. Is this correct? I'm afraid there's something wrong and I'm not seeing it.

I am reading Stillwell's Numbers and Geometry. There is an exercise aboue egyptian fractions which is the following:

I've tried to do it in the following way - Expressing an arbitrary fraction $\frac{n}{m}$ as the sum of two aribtrary egipcian fractions:

$$\frac{1}{a}+\frac{1}{b}=\frac{n}{m}$$

$$\frac{1}{a}+\frac{1}{b}-\frac{n}{m}=0$$

$$\frac{bm+am-abn}{abm}=0$$

Then solving for $a$ yields:

$$a=\frac{b m}{b n-m}$$

And then expressing it as a function:

$$f(b)= \frac{b m}{b n-m}$$

I have tested with $\frac{n}{m}=\frac{3}{4}$ on Mathematica and all the sums of unit fractions I've obtained equals $\frac{3}{4}$. Is this correct? I'm afraid there's something wrong and I'm not seeing it.