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 .
- Diophantine sets and the integers (cartesianproduct.wordpress.com)
This is not some great revelation, but it interested me, so might interest some readers (I got it from The Honors Class: Hilbert’s Problems and Their Solvers).
A diophantine equation is one of the form and a diophantine set is a set of numbers to solve a diophantine equation (Hilbert’s 10th problem was to find an algorithm to solve these in the general case – a task we now know to be impossible).
One diophantine set is the integers – which are a solution to
and so on…
- Hilbert’s 10.5th Problem (rjlipton.wordpress.com)
- On the number of solutions to 4/p = 1/n_1 + 1/n_2 + 1/n_3 (terrytao.wordpress.com)
- The Omega Man (beatsnpeace.wordpress.com)
- Counting the number of solutions to the Erdös-Straus equation on unit fractions (terrytao.wordpress.com)
- Hilbert’s seventh problem, and powers of 2 and 3 (terrytao.wordpress.com)