AbstractsMathematics

Let x, y, z, n, α ∈ ℤ with α ≥ 1, p and n ≥ 5 primes. In 2011, Michael Bennett, Florian Luca and Jamie Mulholland showed that the equation involving a twisted sum of cubes [equation omitted] has no pairwise coprime nonzero integer solutions p ≥ 5,n ≥ p²p and p ∉ S where S is the set of primes q for which there exists an elliptic curve of conductor NE ∈ {18q,36q,72q} with at least one nontrivial rational 2-torsion point. In this dissertation, I present a solution that extends the result to include a subset of the primes in S; those q ∈ S for which all curves with conductor NE ∈ {18q,36q,72q} with nontrivial rational 2-torsion have discriminants not of the form ℓ² or -3m² with ℓ,m ∈ ℤ. Using a similar approach, I will classify certain integer solutions to the equation of a twisted sum of fifth powers [equation omitted] which in part generalizes work done from Billerey and Dieulefait in 2009. I will also discuss limitations of the methods for these equations and as they extend to further prime exponents.