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Search: All articles in the CMB digital archive with keyword arithmetic progression

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1. CMB 2014 (vol 57 pp. 551)

Kane, Daniel M.; Kominers, Scott Duke
Asymptotic Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions
For relatively prime positive integers $u_0$ and $r$, we consider the least common multiple $L_n:=\mathop{\textrm{lcm}}(u_0,u_1,\dots, u_n)$ of the finite arithmetic progression $\{u_k:=u_0+kr\}_{k=0}^n$. We derive new lower bounds on $L_n$ that improve upon those obtained previously when either $u_0$ or $n$ is large. When $r$ is prime, our best bound is sharp up to a factor of $n+1$ for $u_0$ properly chosen, and is also nearly sharp as $n\to\infty$.

Keywords:least common multiple, arithmetic progression
Category:11A05

2. CMB 2011 (vol 55 pp. 193)

Ulas, Maciej
Rational Points in Arithmetic Progressions on $y^2=x^n+k$
Let $C$ be a hyperelliptic curve given by the equation $y^2=f(x)$ for $f\in\mathbb{Z}[x]$ without multiple roots. We say that points $P_{i}=(x_{i}, y_{i})\in C(\mathbb{Q})$ for $i=1,2,\dots, m$ are in arithmetic progression if the numbers $x_{i}$ for $i=1,2,\dots, m$ are in arithmetic progression. In this paper we show that there exists a polynomial $k\in\mathbb{Z}[t]$ with the property that on the elliptic curve $\mathcal{E}': y^2=x^3+k(t)$ (defined over the field $\mathbb{Q}(t)$) we can find four points in arithmetic progression that are independent in the group of all $\mathbb{Q}(t)$-rational points on the curve $\mathcal{E}'$. In particular this result generalizes earlier results of Lee and V\'{e}lez. We also show that if $n\in\mathbb{N}$ is odd, then there are infinitely many $k$'s with the property that on curves $y^2=x^n+k$ there are four rational points in arithmetic progressions. In the case when $n$ is even we can find infinitely many $k$'s such that on curves $y^2=x^n+k$ there are six rational points in arithmetic progression.

Keywords:arithmetic progressions, elliptic curves, rational points on hyperelliptic curves
Category:11G05

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