The index of a genus one curve X over a field K
is the smallest degree of an extension
L of K such that X(L) is nonempty.
Let K be a number field.
We prove that for every integer r not divisible
by 8, there is a genus one curve X over K of index r.
Our proof involves an analysis of Kolyvagin's Euler system of Heegner points
combined with explicit computations on the modular curve X0(17).