arXiv subjects

Li-Yuan Wang

Publications and source records attributed to Li-Yuan Wang.

21 records · Page 2Linked to original sources

arXiv2019

On practical numbers of some special forms

In this paper we study practical numbers of some special forms. For any integers $b\ge0$ and $c>0$, we show that if $n^2+bn+c$ is practical for some integer $n>1$, then there are infinitely many nonnegative integers $n$ with $n^2+bn+c$ practical. We also prove that there are infinitely many practical numbers of the form $q^4+2$ with $q$ practical, and that there are infinitely many practical Pythagorean triples $(a,b,c)$ with $\gcd(a,b,c)=6$ (or $\gcd(a,b,c)=4$).

math.NT↗

arXiv2019

Applications of Lerch's theorem and permutations concerning quadratic residues

Let $p$ be an odd prime. For each integer $a$ with $p\nmid a$, the famous Zolotarev's Lemma says that the Legendre symbol $(\frac{a}{p})$ is the sign of the permutation of $\Z/p\Z$ induced by multiplication by $a$. The extension of Zolotarev's result to the case of odd integers was shown by Frobenius. After that, Lerch extended these to all positive integers. In this paper we explore some applications of Lerch's result. For instance, we study permutations involving arbitrary $k$-th power residue modulo $p$ and primitive roots of a power of $p$. Finally, we discuss some permutation problems concerning quadratic residues modulo $p$. In particular, we confirm some conjectures posed by Sun.

math.NT↗

arXiv2018

Some permutations over ${\mathbb F}_p$ concerning primitive roots

Let $p$ be an odd prime and let ${\mathbb F}_p$ denote the finite field with $p$ elements. Suppose that $g$ is a primitive root of ${\mathbb F}_p$. Define the permutation $τ_g:\,{\mathcal H}_p\to{\mathcal H}_p$ by $$ τ_g(b):=\begin{cases} g^b,&\text{if }g^b\in{\mathcal H}_p,\\ -g^b,&\text{if }g^b\not\in{\mathcal H}_p,\\ \end{cases} $$ for each $b\in{\mathcal H}_p$, where ${\mathcal H}_p=\{1,2,\ldots,(p-1)/2\}$ is viewed as a subset of ${\mathbb F}_p$. In this paper, we investigate the sign of $τ_g$. For example, if $p\equiv 5\pmod{8}$, then $$ (-1)^{|τ_g|}=(-1)^{\frac{1}{4}(h(-4p)+2)} $$ for every primitive root $g$, where $h(-4p)$ is the class number of the imaginary quadratic field ${\mathbb Q}(\sqrt{-4p})$.

math.NT↗