Unknotting number is not additive under connected sum

We give the first examples of a pair of knots $K_1$,$K_2$ in the 3-sphere for which their unknotting numbers satisfy $u(K_1\mathrm{\#}{K}_2)<u(K_1)+u(K_2)$ . This answers question 1.69(B) from Kirby’s problem list, “Problems in low-dimensional topology”, in the negative.