Thomson's problem, dating from 1904, asks which configurations of N points on the 2-sphere minimize the total electrostatic potential.   This problem and its generalizations have a huge literature.  The most common generalization is to replace the electrostatic potential with a general power-law  potential.   (The electrostatic potential is the power-law potential with exponent 1.).  The case N=5 has been notoriously difficult, even for the electrostatic potential. In the late 70s,  the  physicists Melnyk, Knopf, and Smith observed experimentally that for the case N=5, the minimizing configuration is the triangular bi-pyramid for all power law exponents up to some critical value 15.048077..., and then the answer changes to a family of pyramids with square bases. In this talk I'll explain my  computer-assisted proof of  this conjecture, up to exponent 15+25/512=15.04809... My result contains Thomson's original 5-electron problem as a special case.