Parousia Rockstroh

Harvey Mudd College Mathematics 2008

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Final Thesis Report: Properties of the Golomb Topology and Fibonacci Cycles under Integer Moduli
Thesis Advisor: Prof. Francis Su
Second Reader: Prof. Sanjai Gupta

Properties of the Golomb Topology and Fibonacci Cycles under Integer Moduli

In this thesis we study both the Golomb topology and the Fibonacci sequence. The first chapter of the thesis is devoted to exploring the topological structure o f the Golomb topology in addition to some striking number theoretic applications . In particular, we use the topology to show that there are infinitely many prim es, and we also prove that Dirichlet's theorem on primes in arithmetic progressi ons is equivalent to stating that the primes are dense in the Golomb topology. F or the remainder of the thesis, we explore the idea of the density of a sequence in the Golomb topology by embedding a well known sequence - the Fibonacci seque nce - in the topology. This leads to an exploration of some of the properties of the Fibonacci sequence, including a cyclic structure that the sequence exhibits when reduced modulo an integer. In the final chapter, we place a bound on the c ycle length for prime moduli by appealing to techniques from algebra.