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We investigate Benford's law in relation to fractal geometry. Basic fractals, such as the Cantor set and Sierpinski triangle are obtained as the limit of iterative sets, and the unique measures of their components follow a geometric distribution, which is Benford in most bases. Building on this intuition, we aim to study this distribution in more complicated fractals. We examine the Laurent coefficients of a Riemann mapping and the Taylor coefficients of its reciprocal function from the exterior of the Mandelbrot set to the complement of the unit disk. These coefficients are 2-adic rational numbers, and through statistical testing, we demonstrate that the numerators and denominators are a good fit for Benford's law. We offer additional conjectures and observations about these coefficients. In particular, we highlight certain arithmetic subsequences related to the coefficients' denominators, provide an estimate for their slope, and describe efficient methods to compute them., Comment: Updated to fix typographical errors and improve readability. Updated graphs and explanations to make them more concise and readable, and reworded interpretations of testing to make them more appropriate. Results remain unchanged from the version submitted to Fractal and Fractional for publication

Two well-studied Diophantine equations are those of Pythagorean triples and elliptic curves, for the first we have a parametrization through rational points on the unit circle, and for the second we have a structure theorem for the group of rational solutions. Recently, Yekutieli discussed a connection between these two problems, and described the group structure of Pythagorean triples and the number of triples for a given hypotenuse. In arXiv:2112.03663 we generalized these methods and results to Pell's equation. We find a similar group structure and count on the number of solutions for a given $z$ to $x^2 + Dy^2 = z^2$ when $D$ is 1 or 2 modulo 4 and the class group of $\mathbb{Q}[\sqrt{-D}]$ is a free $\mathbb{Z}_2$ module, which always happens if the class number is at most 2. In this paper, we discuss the main results of arXiv:2112.03663 using some concrete examples in the case of $D=105$., Comment: 15 pages, 5 tables

Two well-studied Diophantine equations are those of Pythagorean triples and elliptic curves; for the first, we have a parametrization through rational points on the unit circle, and for the second we have a structure theorem for the group of rational solutions. Recently Yekutieli discussed a connection between these two problems and described the group structure of Pythagorean triples and the number of triples for a given hypotenuse. We generalize these methods and results to Pell's equation. We find a similar group structure and count on the number of solutions for a given $z$ to $x^2 + Dy^2 = z^2$ when $D$ is 1 or 2 modulo 4 and the class group of $\mathbb{Q}[\sqrt{-D}]$ is a free $\mathbb{Z}_2$ module, which always happens if the class number is at most 2. We give examples of when the results hold for a class number greater than 2, as well as an example with different behavior when the class group does not have this structure., Comment: 13 pages, 1 figure

Zeckendorf's theorem states that every positive integer can be written uniquely as the sum of non-consecutive shifted Fibonacci numbers $\{F_n\}$, where we take $F_1=1$ and $F_2=2$. This has been generalized for any Positive Linear Recurrence Sequence (PLRS), which informally is a sequence satisfying a homogeneous linear recurrence with a positive leading coefficient and non-negative integer coefficients. In this and the preceding paper we provide two approaches to investigate linear recurrences with leading coefficient zero, followed by non-negative integer coefficients, with differences between indices relatively prime (abbreviated ZLRR), via two different approaches. The first approach involves generalizing the definition of a legal decomposition for a PLRS found in Kolo\u{g}lu, Kopp, Miller and Wang. We prove that every positive integer $N$ has a legal decomposition for any ZLRR using the greedy algorithm. We also show that a specific family of ZLRRs lost uniqueness of decompositions. The second approach converts a ZLRR to a PLRR that has the same growth rate. We develop the Zeroing Algorithm, a powerful helper tool for analyzing the behavior of linear recurrence sequences. We use it to prove a very general result that guarantees the possibility of conversion between certain recurrences, and develop a method to quickly determine whether a sequence diverges to $+\infty$ or $-\infty$, given any real initial values. This paper investigates the second approach., Comment: 24 pages, 1 photo. Portions of this work previously appeared as earlier versions of arXiv:2001.08455 which has been split for publication. This version includes a new author and new results on the run-time of the Zeroing Algorithm. Final version, to appear in the Fibonacci Quarterly

While there are many identities involving the Euler and Bernoulli numbers, they are usually proved analytically or inductively. We prove two identities involving Euler and Bernoulli numbers with combinatorial reasoning via up-down permutations., Comment: 3 pages

The Mordell-Weil groups $E(\mathbb{Q})$ of elliptic curves influence the structures of their quadratic twists $E_{-D}(\mathbb{Q})$ and the ideal class groups $\mathrm{CL}(-D)$ of imaginary quadratic fields. For appropriate $(u,v) \in \mathbb{Z}^2$, we define a family of homomorphisms $\Phi_{u,v}: E(\mathbb{Q}) \rightarrow \mathrm{CL}(-D)$ for particular negative fundamental discriminants $-D:=-D_E(u,v)$, which we use to simultaneously address questions related to lower bounds for class numbers, the structures of class groups, and ranks of quadratic twists. Specifically, given an elliptic curve $E$ of rank $r$, let $\Psi_E$ be the set of suitable fundamental discriminants $-D<0$ satisfying the following three conditions: the quadratic twist $E_{-D}$ has rank at least 1; $E_{\text{tor}}(\mathbb{Q})$ is a subgroup of $\mathrm{CL}(-D)$; and $h(-D)$ satisfies an effective lower bound which grows asymptotically like $c(E) \log (D)^{\frac{r}{2}}$ as $D \to \infty$. Then for any $\varepsilon > 0$, we show that as $X \to \infty$, we have $$\#\, \left\{-X < -D < 0: -D \in \Psi_E \right \} \, \gg_{\varepsilon} X^{\frac{1}{2}-\varepsilon}.$$ In particular, if $\ell \in \{3,5,7\}$ and $\ell \mid |E_{\mathrm{tor}}(\mathbb{Q})|$, then the number of such discriminants $-D$ for which $\ell \mid h(-D)$ is $\gg_{\varepsilon} X^{\frac{1}{2}-\varepsilon}.$ Moreover, assuming the Parity Conjecture, our results hold with the additional condition that the quadratic twist $E_{-D}$ has rank at least 2., Comment: 17 pages, 1 table

Given a finite set of integers $A$, its sumset is $A+A:= \{a_i+a_j \mid a_i,a_j\in A\}$. We examine $|A+A|$ as a random variable, where $A\subset I_n = [0,n-1]$, the set of integers from 0 to $n-1$, so that each element of $I_n$ is in $A$ with a fixed probability $p \in (0,1)$. Recently, Martin and O'Bryant studied the case in which $p=1/2$ and found a closed form for $\mathbb{E}[|A+A|]$. Lazarev, Miller, and O'Bryant extended the result to find a numerical estimate for $\text{Var}(|A+A|)$ and bounds on the number of missing sums in $A+A$, $m_{n\,;\,p}(k) := \mathbb{P}(2n-1-|A+A|=k)$. Their primary tool was a graph-theoretic framework which we now generalize to provide a closed form for $\mathbb{E}[|A+A|]$ and $\text{Var}(|A+A|)$ for all $p\in (0,1)$ and establish good bounds for $\mathbb{E}[|A+A|]$ and $m_{n\,;\,p}(k)$. We continue to investigate $m_{n\,;\,p}(k)$ by studying $m_p(k) = \lim_{n\to\infty}m_{n\,;\,p}(k)$, proven to exist by Zhao. Lazarev, Miller, and O'Bryant proved that, for $p=1/2$, $m_{1/2}(6)>m_{1/2}(7)m_{p}(1)