arXiv subjects
Publications and source records attributed to Haojie Hong.
We investigate the monogenity of irreducible factors of the Fibonacci polynomials $F_n(x)$ and the Lucas polynomials $L_n(x)$. Our main results show that for every odd positive integer $n$, all irreducible factors of $F_n(x)$ are monogenic, and for every even positive integer $n$, all irreducible factors of $L_n(x)$ are monogenic.
In this note, we prove that for any given positive integer $κ$, when $n$ is bigger than a constant explicitly depending on $κ$, the $n$-th Fibonacci number has a primitive divisor not less than $(κ+1)n-1$.
Stewart (2013) proved that the biggest prime divisor of the $n$th term of a Lucas sequence of integers grows quicker than $n$, answering famous questions of Erdős and Schinzel. In this note we obtain a fully explicit and, in a sense, uniform version of Stewart's result.
Let $γ$ be an algebraic number of degree $2$ and not a root of unity. In this note we show that there exists a prime ideal $\mathfrak{p}$ of $\mathbb{Q}(γ)$ satisfying $ν_\mathfrak{p}(γ^n-1)\ge 1$, such that the rational prime $p$ underlying $\mathfrak{p}$ grows quicker than $n$.
Let $E$ be an elliptic curve over the finite field $\mathbb F_q$. We prove that, when $n$ is a sufficiently large positive integer, $\#E(\mathbb F_{q^n})$ has a prime factor exceeding $n\exp(c\log n/\log\log n)$.