arXiv subjects
Publications and source records attributed to Francesco Pappalardi.
We exhibit a new application of two dimensional covering systems, examples of integer pairs $a,b$ for which $a^m-b^n$ has a prime divisor from some given finite set of primes, for every pair of integers $m,n\geq 0$. This leads us to conjecture what are the only possible obstructions to $|a^m-b^n|$ taking on infinitely many distinct prime values.
Under GRH, any element in the multiplicative group of a number field $K$ that is globally primitive (i.e., not a perfect power in $K^*$) is a primitive root modulo a set of primes of $K$ of positive density. For elliptic curves $E/K$ that are known to have infinitely many primes $\mathfrak p$ of cyclic reduction, possibly under GRH, a globally primitive point $P\in E(K)$ may fail to generate any of the point groups $E(k_{\mathfrak p})$. We describe this phenomenon in terms of an associated Galois representation $ρ_{E/K, P}:G_K\to\mathrm{GL}_3(\hat{\mathbf Z})$, and use it to construct non-trivial examples of global points on elliptic curves that are locally imprimitive.
For a given finitely generated multiplicative subgroup of the rationals which possibly contain negative numbers, we derive, subject to GRH, formulas for the densities of primes for which the index of the reduction group has a given value. We completely classify the cases of rank one torsion groups for which the density vanishes and the the set of primes for which the index of the reduction group has a given value, is finite. For higher rank groups we propose some partial results. Finally, we propose some computations of examples comparing the approximated density computed with primes up to $10^{10}$ and that predicted by the Riemann Hypothesis.
Given a positive integer $n$ we let $A_k(n)$ be the number of positive integers $a$ such that $\frac{a}{n}=\frac{1}{m_1}+\frac{1}{m_2}+\cdots+\frac{1}{m_k}$ for some $m_1,m_2,\ldots,m_k\in {\mathbb N}$. We show that $x(\log x)^3\ll \sum_{p\le x} A_3(p)\ll x(\log x)^5$ as $x\rightarrow\infty$.
In this paper, we give several asymptotic formulas for the number of multiplicatively dependent vectors of algebraic numbers of fixed degree, or within a fixed number field, and bounded height.
We study the collection of group structures that can be realized as a group of rational points on an elliptic curve over a finite field (such groups are well known to be of rank at most two). We also study various subsets of this collection which correspond to curves over prime fields or to curves with a prescribed torsion. Some of our results are rigorous and are based on recent advances in analytic number theory, some are conditional under certain widely believed conjectures, and others are purely heuristic in nature.
We remark on pseudo-elliptic integrals and on exceptional function fields, namely function fields defined over an infinite base field but nonetheless containing non-trivial units. Our emphasis is on some elementary criteria that must be satisfied by a squarefree polynomial whose square root generates a quadratic function field with non-trivial unit. We detail the genus 1 case.
We prove an asymptotic formula for the number of permutation for which the associated permutation polynomial has degree smaller than $q-2$.