Prakod Ngamlamai

Thesis Advisor: Dr. Peter Kagey (Harvey Mudd College - Department of Mathematics)

Second Reader: Dr. Michael Orrison (Harvey Mudd College - Department of Mathematics)

Contact: pngamlamai@hmc.edu

Generalized Far-Difference Decomposition

Integers are often decomposed into forms $\sum c_i*B^i$ as base B representation where $c_i,B$ are positive integers. Zeckendorf first proved a result of base $\phi$ decomposition by providing the rules for decomposing integers into Fibonacci numbers, which are approximation of powers of $\phI$ the golden ratio. Hannah Alpert further studied decomposition of integers into Fibonacci numbers by introducing negative terms, coining the Far-Difference Representation. Our work aims to generalize her work to more than just Fibonacci numbers and consider other sequences called Positive Linear Recurrence Sequences or PLRS, as had been done with Zeckendorf decomposition. To do so, we describe general rules of the decompositions, such as uniqueness, and attempt to prove other properties such as summand-minimality for certain PLRSs.