## 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).

None yet.

kjell@home.

# Kjell Wooding

Ph.D   M.Sc  B.Eng