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I'm a huge fan of conjectures, and I'm fascinated by this new one. The Erdős-Strauss conjecture is that $\frac{4}{n} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$, where $n$ is greater than or equal to $2$ and is a natural number. Or, $\frac{4}{n}$, where $n$ is greater or equal to two and is a natural number, can be written as the sum of three unit fractions.
I know that I can eliminate all even numbers, because $\frac{4}{n}$ where $n$ is even can be written as $\frac{1}{0.5n} + \frac{1}{n} + \frac{1}{n} = \frac{2}{n} + \frac{1}{n} + \frac{1}{n} = \frac{4}{n}$. The way I test the odd numbers is I subtract the nearest unit fraction less than $\frac{4}{n}$ and see if the difference can be written as the sum of two unit fractions. But as $n$ keeps getting higher, my strategy doesn't work for long.
So, my question is, what's a better strategy?

oernik
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Mathemagician314
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  • I do not think that we can do much better in general. Since the conjecture is still open, a general representation cannot be known.By the way, if $n$ is even, we can divide numerator and denominator by $2$ having the same fraction and an abvious representation with only $2$ fractions. The conjecture has been checked upto a very high limit. – Peter Jan 24 '20 at 12:50
  • The search limit is apparently $n=10^{17}$ – Peter Jan 24 '20 at 13:17

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