Classical and Modular Approaches to Exponential Diophantine Equations I. FibonacciFibonacci and Lucas Perfect Powers
Modular forms are used to demonstrate that the only perfect powers in the Fibonacci sequence are 0, 1, 8 and 144 and the only perfect powers in the Lucas sequence are 1 and 4.Classical and modular approaches to exponential Diophantine equations II. TheThe Lebesgue-Nagell equation
Modular forms are used to solve the Lebesgue-Nagell equation.Modular magicDensest sphere packing
Modular forms find the lattice with the densest sphere packing problem in dimensions 8 ($E_8$ lattice) and in dimension 24 (Leech lattice).Ramanujan's constant
Modular forms explain why $e^{\pi\sqrt{163}}$ is so close to an integer.
