Amir Ghadermarzi

• PhD
• University of British Columbia

Let $P$ be a non torsion integral point on the minimal 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...

Let (a, b, c) be a primitive Pythagorean triple. Set a=m2-n2, b=2mn , and c=m2+n2 with m and n positive coprime integers, m>n and m≢n(mod2). A famous conjecture of Jeśmanowicz asserts that the only positive integer solution to the Diophantine equation ax+by=cz is (x,y,z)=(2,2,2). A solution (x,y,z)≠(2,2,2) of this equation is called an exceptional...

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 Je\'{s}manowicz 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 fo...

In this note we find all the solutions to the equation x² + 2 α 3 β 19 γ = y ⁿ in nonnegative unknowns with n ≥ 3 and gcd( x , y ) = 1, and nonnegative solutions to x² + 2 α 3 β 13 γ = y ⁿ with n ≥ 3, gcd( x , y ) = 1, except when α = 0 and x . β . γ is odd.

Let $D$ be a positive nonsquare integer, $p$ a prime number with $p\nmid D$ and $0 . We show that there exist effectively computable constants $C_{1}$ and $C_{2}$ such that if there is a solution to $x^{2}+D=p^{n}$ with $p^{n}>C_{1}$ , then for every $x>C_{2}$ with $x^{2}+D=p^{n}m$ we have $m>x^{\unicode[STIX]{x1D70E}}$ . As an application, we show...

We solve the Diophantine equation $Y^{2}=X^{3}+k$ for all nonzero integers $k$ with $|k|\leqslant 10^{7}$ . 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.

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.

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.