ABC Conjecture
a+b=c, hcf(a, b)=1, c<=C_0rad(abc)
Disproof:
a=1
c<=C_0rad(abc) = C_0rad((c-1)c)
[let c->999999...]
->C_0rad(c^2) = C_0rad(c)
But c<=C_0rad(c) - clearly no bounds on C_0.