On Arithmetic Progressions and Perfect Powers

In this thesis we will consider the problems that occur at the intersection
of arithmetic progressions and perfect powers. In particular we will study
the Erd˝os-Selfridge curves, By^l = x(x + d). . .(x + (k − 1)d), and sums of
powers of arithmetic progressions, in particular y
^l = (x−d)^3+x^3+ (x+d)^3
.
We shall study these curves using aspects of algebraic and analytic number
theory. To all the equations studied we shall show that a putative solution
gives rise to solutions of (potentially many) Fermat equations. In the case
of Erd˝os-Selfridge curves we will use the modular method to understand the
prime divisors of d for large `. Then we shall attach Dirichlet characters to
such solutions, which allows us to use analytic methods regarding bounds
on the value of sums of characters. These bounds will allow us to show that
there can’t be too many simultaneous solutions to the Fermat equations
we described. This leads to a contradiction for large k, as the number of
Fermat equations generated will grow faster than the possible number of
simultaneous solutions.
We study the arithmetic progression curves by attaching Fermat equations
of signature (l, l, 2). We then use the classical modular method to attach
Frey-Hellegouarch curves and level lowered modular forms. It is possible to
show that the Frey-Hellegouarch curves that associate to modular forms in
a non-trivial cuspidal newspace are all quadratic twists of each other. It is
then possible to compute if there are modular forms of the right level that
could associate to such a twist of an elliptic curve.