Simultaneous prime values of binary forms

The twin prime conjecture asserts that there are infinitely many positive integers x such that x and x+2 are simultaneously prime. In this thesis, we consider a two-variable analogue of this problem. Let F(x, y) be a positive definite quadratic form and G(x, y) a linear form, both with integer coefficients. Suppose for any prime p there exist l, m such that p t F(l, m)G(l, m). Then we prove that there are infinitely many l, m in Z such that both F(l, m), G(l, m) are primes. In fact, our proof extends to primes in arithmetic progressions F(l, m)= a mod q and G(l, m)=b mod q. The main result (when q=1) was first obtained independently by the author and another team of researchers D. Schindler and S. Y. Xiao. The extension is joint work with them.