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UBC Theses and Dissertations

Equations in the primes Cook, Brian Michael


We provide results related to the study of prime points on level sets of homogeneous integral forms which are linear or quadratic. In the linear case we present an extension of the Green-Tao Theorem, which finds affine copies of finite intervals in relatively dense subsets of the primes, to a higher dimensional setting in which one finds affine copies of suitably generic point configurations in relatively dense subsets of a Cartesian product of the primes. For general integral quadratic forms we present a result which is a Birch-Goldbach type theorem for a single quadratic form with sufficient rank. This guarantees solubility among the primes on the level set of a quadratic form subject to local conditions. This is an extension of a well known result of Hua.

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