arXiv subjects
Publications and source records attributed to Simon Earp-Lynch.
We examine the equation $y^{a}=\sum\limits_{i=1}^{k}F_{n_{i}}$ for positive integers $y,a\geq 2$ and $k\geq3$. This equation can be expressed as a problem in terms of the Zeckendorf representations of integers. Using bounds on linear forms in logarithms and Baker-Davenport reduction methods, we are able to completely solve the equation for $ y\leq 25000$ when $k=3$, for $y\leq 1000$ when $k= 4$, for $y\leq 40$ when $k= 5$, and for $y\leq 3$ when $k=6$.
A set of $m$ distinct positive integers $\{a_{1},\dots a_{m}\}$ is called a $D(q)$-$m$-tuple for nonzero integer $q$ if the product of any two increased by $q$, $a_{i}a_{j}+q$, $i\neq j$ is a perfect square. Due to certain properties of the sequence, there are many $D(q)$-Diophantine triples related to the Fibonacci numbers. A result of Ba\'{c}i\'{c} and Filipin characterizes the solutions of Pellian equations that correspond to $D(4)$-Diophantine triples of a certain form. We generalize this result in order to characterize the solutions of Pellian equations that correspond to $D(l^2)$-Diophantine triples satisfying particular divisibility conditions. % Subsequently, we employ this result and bounds on linear forms in logarithms of algebraic numbers in order to classify all $D(9)$ and $D(64)$-Diophantine triples of the form $\{F_{2n+8},9F_{2n+4},F_{k}\}$ and $\{F_{2n+12},16F_{2n+6},F_{k}\}$, where $F_{i}$ denotes the $i$th Fibonacci number.
We solve the equation $\sum\limits_{j=1}^{k}jU_{j}(x,y)^{p}=U_{n}(x,y)^{q}$ positive integers $x,p,q,k,n$, with $y=\pm1$ and $\max\{p,q\}\leq11$, where $U_{m}(x,y)=\frac{\alpha^{m}-\beta^{m}}{\alpha-\beta}$ for $\alpha$ and $\beta$ roots of the polynomial $t^2-xt+y$. This generalizes existing results on similar equations, wherein the sequence was fixed as either the Fibonacci or Pell numbers. In addition, we find all solutions with $k=2$ and $y=\pm1$.
We show that Mordell curves with arithmetic progressions in the $y$-coordinate of length $7$ have not been ruled out by previous work, and we give non-isomorphic families of Mordell curves with $y$-arithmetic progressions of length $6$. We also construct parametric families of elliptic curves of moderate rank, with subfamilies possessing rational points in arithmetic progression.