## Ph.D. Research

### The Sieve Problem in One- and Two-Dimensions

My Ph.D. Research is concerned with the development of tools and techniques for solving systems of simultaneous congruences—a.k.a the congruential sieve problem—in both one- and two-dimensions.

Though many problems in number theory can be reduced to an instance of
the congruential sieve problem, one problem in particular—that of
*primality proving*—is examined in detail.

In previous work, we provided numerical evidence for a conjecture that primality may be proved with complexity $(\log N)^{3+o(1)}$ using quantities known as pseudosquares and pseudocubes. My Ph.D. thesis examines an alternate definition of pseudocube—the Eisenstein pseudocube—which leads to a more efficient primality proving method for primes $p \equiv 1 \pmod 3$. In particular, I:

- develop the notion of an Eisenstein pseudocube, and an associated primality proving algorithm;
- reduce the problem of finding Eisenstein pseudocubes to an instance of the two-dimensional sieve problem;
- extend the CASSIE toolkit to solve instances of a two-dimensional sieve problem;
- develop a general-purpose hardware framework for implementing custom computing devices on Xilinx FPGA devices;
- design and implement FPGA-based sieve device using this framework;
- evaluate the performance of the prototype hardware for solving two-dimensional sieve problems; and
- enumerate Eisenstein pseudocubes using these tools.

Kjell Wooding. The Sieve Problem in One- and Two-Dimensions. Ph.D. Thesis. University of Calgary (2010).