More on the Erdős–Straus conjecture

Number of ways to write an even number n as th...
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As a reminder, the Erdős–Straus conjecture is: \forall n \in \mathbb{N}_1:\frac{4}{n+3}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} .

Let {n+3} = k , we can see if k is even, then k=2j and \frac{4}{k} = \frac{4}{2j} = \frac{2}{j} =\frac{1}{2j} + \frac{1}{2j} + \frac{1}{j} so the conjecture holds for all even k . In fact this holds for k=2, j=1 also.

For odds it’s not so simple, though an expansion does exist for the odd numbers of the form k=4j+3


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