#### Date of Award

Summer 2021

#### Document Type

Open Access Dissertation

#### Department

Mathematics

#### First Advisor

Michael Filaseta

#### Abstract

This dissertation considers three different topics. In the first part of the dissertation, we use Newton Polygons to show that for the arithmetic functions g(n) = n t , where t ≥ 1 is an integer, the polynomials defined with initial condition P g 0 (X) = 1 and recursion P g n (X) = X n Xn k=1 g(k)P g n−k (X) are X/ (n!) times an irreducible polynomial. In the second part of the dissertation, we show that, for 3 ≤ n ≤ 8, there are infinitely many 2-adic integer solutions to the Prouhet-Tarry-Escott (PTE) problem, that are not rational integer solutions. In particular, we look at the 2-adic valuation of a certain constant associated with the PTE problem and for the case n = 8 there exist solutions whose valuation is strictly less than any known rational integer solution. In the third part of the dissertation, we obtain a number of results pertaining to polynomials f(x) with non-negative integer coefficients that take on a prime value at x = b, where b ≥ 2 is an integer. In particular, we give an explicit bound M1(b) such that if the coefficients of f(x) are each ≤ M1(b), then f(x) is irreducible. We also show that there are similarly explicit bounds M2(b), M3(b) and M4(b), for b sufficiently large (made explicit), that can be placed on the coefficients of f(x) such that if f(x) is reducible then it must be divisible by at least one of the shifted cyclotomic polynomials Φ3(x − b), Φ4(x − b) or Φ6(x − b).

#### Recommended Citation

Foster, J. C.(2021). *Polynomials, Primes and the PTE Problem.* (Doctoral dissertation). Retrieved from https://scholarcommons.sc.edu/etd/6407