Diophantine equations involving factorials and lattice points close to smooth curves

By "Diophantine equation" we merely mean an equation in which we are only concerned with positive integer solutions. Hence, the goal is to completely solve a particular Diophantine equation(s). When referring to a lattice point, we will mean an integer coordinate point lying in the plane.;The focus is on two main topics, Diophantine equations involving factorials and counting lattice points close to smooth curves. The first topic possess some outstanding problems aging over 130 years. The main results are improvements of a problem of Florian Luca. The second topic has rudimentary roots in numerous other subjects including approximation theory, combinatorics, complex analysis, and geometry. We provide an introduction and history of some recent applications of this method. Our work is for curves in two dimensions, and in general a smorgasbord of problems have erupted from this topic. The improvements given in this paper are followed with an application to a problem regarding Fq -rational points on curves of small genus over the finite field Fq.