I came across this as a result of links to stories about Hilbert’s tenth problem, and it looks fun, so I thought I’d write a little about it.

The Erdős–Straus conjecture is that for any integer then where , , and are positive integers.

This is equivalent to a diophantine equation:

Which is, apparently, trivially solvable for composite (non-prime) numbers. And we can obviously see that if then if we had an expansion for then the expansion for would simply be – so if we found the conjecture true for a prime then it would be true for all that prime’s multiples. Hence to disprove the conjecture one needs to find a prime for which it is false.

And, indeed, the truth of the conjecture is an open question, though computationally it has been verified as true up to .

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- Diophantine sets and the integers (cartesianproduct.wordpress.com)