Some problems in multiplicative number theory
Junsoo Ha · 2014
Let q be a power of a prime p. In the first part of this thesis, we establish the upper bound of the least prime primitive root mod q by p^3.1. We say a polynomial in F_q [T] is m-smooth if all of its irreducible factors are of degree less than or equal to m. Let N(n, m) be the number of solutions to the polynomial equation X+Y=2Z where all variables are m-smooth polynomials of degree n. In the second part of this thesis, we establish a lower bound on N(n, m) when (8+d) log_q n < = m < = n^1/2 for small d, and prove the analog of the xyz conjecture of Lagarias and Soundararajan in the polynomial rings over finite fields.