arXiv subjects

Amir Ghadermarzi

Publications and source records attributed to Amir Ghadermarzi.

5 recordsLinked to original sources

arXiv2023

Multiples of integral points on Mordell curves

Let $B$ be a sixth-power-free integer and $P$ be a non-torsion point on the Mordell curve $E_B:y^2=x^3+B$. In this paper, we study integral multiples $[n]P$ of $P$. Among other results, we show that $P$ has at most three integral multiples with $n>1$. This result is sharp in the sense that there are points $P$ with exactly three integral multiples $[n]P$ and $n>1$. As an application, we discuss the number of integral points on the quasi-minimal model of rank 1 Mordell curves.

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arXiv2020

On The Exceptional solutions of Jeśmanowicz' conjecture

Let $(a,b,c)$ be a primitive Pythagorean triple. Set $a=m^2-n^2$,$b=2mn$, and $c=m^2+n^2$ with $m$ and $n$ positive coprime integers, $m>n $ and $ m \not \equiv n \pmod 2$. A famous conjecture of Jeśmanowicz asserts that the only positive solution to the Diophantine equation $a^x+b^y=c^z$ is $(x,y,z)(2,2,2).$ In this note, we will prove that for any $n>0$ there exists an explicit constant $c(n)>0$ such that if $m> c(n)$, then the above equation has no exceptional solution when all $x$,$y$ and $z$ are even. Our result improves that of Fu and Yang [11]. As an application, we will show that if $4 \mid\!\mid m$ and $m > c(n)$ Jeśmanowicz' conjecture holds.

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arXiv2018

On the factorization of $x^2+D$

Let $D$ be a positive nonsquare integer, $p$ a prime number with $p \nmid D$, and $0< σ< 0.847$. We show that if the equation $x^2+D=p^n$ has a huge solution $(x_0,n_0)_{(p,σ)}$, then there exists an effectively computable constant $C_p$ such that for every $x> C_P$ with $x^2+D=p^n.m $, we have $ m > x^σ$. As an application, we show that for $x \neq \{1015,5 \}$, if the equation $x^2+76=101^n.m $ holds, we have $ m > x^{0.14}$. .

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arXiv2014

Extremal families of cubic Thue equations

We exactly determine the integral solutions to a previously untreated infinite family of cubic Thue equations of the form $F(x,y)=1$ with at least $5$ such solutions. Our approach combines elementary arguments, with lower bounds for linear forms in logarithms and lattice-basis reduction.

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arXiv2013

Mordell's equation : a classical approach

We solve the Diophantine equation $Y^2=X^3+k$ for all nonzero integers $k$ with $|k| \leq 10^7$. Our approach uses a classical connection between these equations and cubic Thue equations. The latter can be treated algorithmically via lower bounds for linear forms in logarithms in conjunction with lattice-basis reduction.

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