ABC conjecture

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The abc conjecture asserts, roughly speaking, that if a+b=c and a,b,c are coprime, then a,b,c cannot all be too smooth; in particular, the product of all the primes dividing a, b, or c has to exceed [math]c^{1-\varepsilon}[/math] for any fixed [math]\varepsilon \gt 0[/math] (if a,b,c are smooth).

This shows for instance that [math](1-\varepsilon) \log N / 3[/math]-smooth a,b,c of size N which are coprime cannot sum to form a+b=c. This unfortunately seems to be too weak to be of much use for the finding primes project.