Document Type
Honors Thesis
Abstract
Grimm's Conjecture asserts that given any sequence of consecutive natural numbers, say n +1, n + 2,...,n + k for integers n and k. there exists k distinct primes, p1, p2,..., Pk, such that n + i is divisible by Pi, for all i = 1,2,..., k. A construction for graphical repres entations was demonstrated by setting each composite and each prime as a node and then drawing an edge between composite and prime nodes to indicate divisibility. Valuations and related works were used to prove properties of those sub-graphs. Algorithms were then explored to help with finding experimental results regarding minimal interval lengths for counterexamples. It was then shown given a known valuation for a particular node, valuations for many nearby if not all nodes could in turn be evaluated for the same prime. It was then shown Grimm's Conjecture holds for all k ≤ 31 and a new method for deriving a relationship between n and k was found.
Publication Date
5-1-2022
Language
English
License
This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 International License.
Recommended Citation
Hands, Andrew R., "ON GRAPHICAL REPRESENTATIONS OF GRIMM’S CONJECTURE AND MINIMAL INTERVAL LENGTHS" (2022). 2022 Spring Honors Capstone Projects. 46.
https://mavmatrix.uta.edu/honors_spring2022/46