advocated by Shapere and Wilczek [67, 68]. Multi-valued Hamiltonians have appeared in at least two contexts. Most recently, they have resulted from Legendre transforming Lagrangians whose velocity dependence is not convex [67, 68], which invariably leads to a Riemann surface phase-space structure, with m

posted on the arXiv. A revised version is in preparation and nearly complete. The revision updates the references and places additional emphasis on Legendre duality. Additional implicit and explicit solutions are constructed and analyzed in some detail. The (super) twistor formulation, in D=3 and 4, and

oncepts from computational number theory. We will start with algorithms dating back to antiquity (Euclid) and work our way up to Fermat, Euler, and Legendre. We will also mention in passing a few useful concepts from 20th century math. For those seeing this material for the first time I recommend also ta