Unknotting number and connected sums: The knots $4_1$ and $5_1$

We show that the knots $K\in \{4_1,5_1\}$ can be paired with a corresponding knot $K^{\mathrm{\prime }}$ such that $u(K\mathrm{\#}{K}^{\mathrm{\prime }})<u(K)+u(K^{\mathrm{\prime }})$. As a consequence unknotting number fails to be additive for these knots. We also provide a candidate knot $K^{\mathrm{\prime }}$ for the knot $3_1$.