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egyptian fractions

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can all fractions with a numerator of two be written as the sum of two different unitary fractions with mathematical proof

Jan 7, 2020

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egyptian fractions

can all fractions with a numerator of two be written as the sum of two different unitary fractions with mathematical proof

I assume:
$$\dfrac{2}{n}=\dfrac{1}{a}+\dfrac{1}{b}$$

if $$n$$ is even then we set $$n=2k$$:

$$\mathbf{\dfrac{2}{n}}=\dfrac{2}{2k}=\dfrac{1}{k}\mathbf{=\dfrac{1}{k+1} + \dfrac{1}{k(k+1)} }$$

if $$n$$ is odd then we set $$n=2k-1$$:

$$\mathbf{\dfrac{2}{n}}=\dfrac{2}{2k-1}\mathbf{=\dfrac{1}{k} + \dfrac{1}{k(2k-1)} }$$

Jan 7, 2020