Date of Award
Honors College Thesis
Karen Kohl, Ph.D.
The study of p-adic valuations is connected to the problem of factorization of integers, an essential question in number theory and computer science. Given a nonzero integer n and prime number p, the p-adic valuation of n, which is commonly denoted as νp(n), is the greatest non-negative integer ν such that p ν | n. In this paper, we analyze the properties of the 2-adic valuations of some integer sequences constructed from Ulam square spirals. Most sequences considered were diagonal sequences of the form 4n 2 + bn + c from the Ulam spiral with center value of 1. Other sequences related to various Ulam square spirals were selected from the Online Encyclopedia of Integer Sequences (OEIS). Conjectures of the 2-adic valuations of these sequences were made based on observations of the binary tree representations of their valuations. We found explicit closed forms for some sequences with finitely many valuations. When sequences produced infinitely many valuations, these results were proved using an adaptation of the Hensel’s lemma, previously used by Almodovar et al in their study of 2-adic valuation of quadratic polynomials of the form n 2 +a. In both of these cases, we classified a number of similar valuation patterns for the diagonal sequences of Ulam spirals.
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Nguyen, Minh, "2-adic Valuations of Square Spiral Sequences" (2021). Honors Theses. 777.