  We present our study of a set of solutions to the $SU(N)$ Yang-Mills
equations of motion with fractional topological charge. The configurations are
obtained numerically by minimizing the action with gradient flow techniques on
a torus of size $l^2\times(Nl)^2$ with twisted boundary conditions. We pay
special attention to the large $N$ limit, which is taken along a very peculiar
sequence, with the number of colors $N$ and the magnetic flux $m$ selected
respectively as the $n$-th and $n-2$ terms of the Fibonacci sequence. We
discuss the large $N$ scaling of the solutions and analyze several gauge
invariant quantities as the Polyakov loops. We also discuss the so-called
Hamiltonian limit, with one of the large directions sent to infinity, where
these instantons represent tunneling events between inequivalent pure gauge
configurations.
