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arXiv · 2309.16637

The $p$-Adic Valuation Trees for Quadratic Polynomials for Odd Primes

Will Boultinghouse·Emily Hammett·Stephen Hu·Olena Kozhushkina·Rachel Snyder·Justin Trulen
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Abstract

We examine the behavior of the sequences of $p$-adic valuations of quadratic polynomials with integer coefficients for an odd prime $p$ through tree representations. Under this representation, a finite tree corresponds to a periodic sequence, and an infinite tree corresponds to an unbounded sequence. We use the polynomial coefficients to determine whether the $p$-adic valuation trees are finite or infinite, the number of infinite branches, the number of levels, the valuations at terminating nodes, and their relationship to the corresponding sequences.

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Will Boultinghouse, Emily Hammett, Stephen Hu, Olena Kozhushkina, Rachel Snyder, Justin Trulen. 2023-09-28. The $p$-Adic Valuation Trees for Quadratic Polynomials for Odd Primes. https://arxiv.org/abs/2309.16637

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