Primitive roots are one of the most classic object in number theory. A challenging problem on primitive roots is to establish a uniform bound on the size of the least primitive root and the least prime primitive root. In the first part of this talk, I will dicuss on some history and related results and describe my recent result on the uniform bound on prime primitive root.
The Hardy-Littlewood circle method is one of the most powerful tools in the study of additive Diophantine equations. A famous application is due to Vinogradov, who proved that any large odd integer can be written as the sum of primes. Another application is to find a solution to A+B=C, when A, B, C are all smooth numbers, that is, the integers without large prime factors. In the second part of my talk, I will discribe the recent result of Lagarias and Soundararajan, and that their result can be applied to polynomial rings over the finite fields.