ABSTRACT
One must divide one’s time between politics and equations. But our equations are much more important to me. from writings of C. P. Snow in Einstein (1980) M. Goldsmith et al. (eds.)
Albert Einstein (1879-1955) German-born theoretical physicist
In a first course in number theory, elementary Diophantine equations are studied and we assume herein familiarity with the fundamentals such as in [68, Chapters 1, 5, & 7], where norm-form equations, including Pell’s equation, are completely solved via continued fractions, as are linear equations by congruence conditions. We have already encountered some nonlinear Diophantine equations in our developments in Chapter 1, especially in Theorem 1.8 on page 14, where we looked at the Ramanujan-Nagell equation and its solutions. We revisit this equation in §8.2, where we study solutions of the generalized Ramanujan-Nagell equation introduced in Definition 1.10 on page 13. We begin with a theory to solve these latter equations.
