Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSquares in $mathbb{F}_{p^2}$ and permutations involving primitive roots
93
0
0.0
(
0
)
Authors
Hai-Liang Wu
Ask ChatGPT about the research
No Arabic abstract
Let $p=2n+1$ be an odd prime, and let $zeta_{p^2-1}$ be a primitive $(p^2-1)$-th root of unity in the algebraic closure $overline{mathbb{Q}_p}$ of $mathbb{Q}_p$. We let $ginmathbb{Z}_p[zeta_{p^2-1}]$ be a primitive root modulo $pmathbb{Z}_p[zeta_{p^2-1}]$ with $gequiv zeta_{p^2-1}pmod {pmathbb{Z}_p[zeta_{p^2-1}]}$. Let $Deltaequiv3pmod4$ be an arbitrary quadratic non-residue modulo $p$ in $mathbb{Z}$. By the Local Existence Theorem we know that $mathbb{Q}_p(sqrt{Delta})=mathbb{Q}_p(zeta_{p^2-1})$. For all $xinmathbb{Z}[sqrt{Delta}]$ and $yinmathbb{Z}_p[zeta_{p^2-1}]$ we use $bar{x}$ and $bar{y}$ to denote the elements $xmod pmathbb{Z}[sqrt{Delta}]$ and $ymod pmathbb{Z}_p[zeta_{p^2-1}]$ respectively. If we set $a_k=k+sqrt{Delta}$ for $0le kle p-1$, then we can view the sequence $$S := overline{a_0^2}, cdots, overline{a_0^2n^2}, cdots,overline{a_{p-1}^2}, cdots, overline{a_{p-1}^2n^2}cdots, overline{1^2}, cdots,overline{n^2}$$ as a permutation $sigma$ of the sequence $$S^* := overline{g^2}, overline{g^4}, cdots,overline{g^{p^2-1}}.$$ We determine the sign of $sigma$ completely in this paper.
rate research
Read More
In this paper, we determine all the squares in the sequence ${prod_{k=2}^n(k^2-1)}_{n=2}^infty $. From this, one deduces that there are infinitely many squares in this sequence. We also give a formula for the $p$-adic valuation of the terms in this sequence.
Fix $a in mathbb{Z}$, $a otin {0,pm 1}$. A simple argument shows that for each $epsilon > 0$, and almost all (asymptotically 100% of) primes $p$, the multiplicative order of $a$ modulo $p$ exceeds $p^{frac12-epsilon}$. It is an open problem to show the same result with $frac12$ replaced by any larger constant. We show that if $a,b$ are multiplicatively independent, then for almost all primes $p$, one of $a,b,ab, a^2b, ab^2$ has order exceeding $p^{frac{1}{2}+frac{1}{30}}$. The same method allows one to produce, for each $epsilon > 0$, explicit finite sets $mathcal{A}$ with the property that for almost all primes $p$, some element of $mathcal{A}$ has order exceeding $p^{1-epsilon}$. Similar results hold for orders modulo general integers $n$ rather than primes $p$.
In a paper of P. Paillier and J. Villar a conjecture is made about the malleability of an RSA modulus. In this paper we present an explicit algorithm refuting the conjecture. Concretely we can factorize an RSA modulus n using very little information on the factorization of a concrete n coprime to n. However, we believe the conjecture might be true, when imposing some extra conditions on the auxiliary n allowed to be used. In particular, the paper shows how subtle the notion of malleability is.
Given a positive integer $Q$, denote by $mathcal{C}_Q$ the multiplicative cyclic group of order $Q$. Let $n$ be a divisor of $Q$ and $r$ a divisor of $Q/n$. Guided by the well-known formula of Vinogradov for the indicator function of the set of primitive elements of a finite field $mathbb{F}_q$, we derive an expression for the indicator function for the set of $(r,n)$-free elements of $mathcal{C}_Q$, i.e., the subset of the subgroup $mathcal{C}_{Q/n}$ comprising elements that are $r$-free in $mathcal{C}_{Q/n}$, i.e., are not $p$-th powers in $mathcal{C}_{Q/n}$ for any prime $p$ dividing $r$. We deduce a general lower bound for the the number of elements $theta in mathbb{F}_q$ for which $f(theta)$ is $(r,n)$-free and $F(theta)$ is $(R,N)$-free, where $f, F inmathbb{F}_q[x]$ and $n,N$ are divisors of $q-1$ with $rmid (q-1)/n$, $Rmid (q-1)/N$. As an application, we consider the existence of $mathbb{F}_q$-primitive points (i.e., points whose coordinates are primitive elements) on curves like $y^n=f(x)$. In particular, elliptic curves $y^2=f(x)$, where $f$ is a square-free cubic, are studied. We find, for example, all the odd prime powers $q$ for which the elliptic curves $y^2=x^3 pm x$ contain an $mathbb{F}_q$-primitive point.
Let $p>3$ be a prime. Gauss first introduced the polynomial $S_p(x)=prod_{c}(x-zeta_p^c),$ where $0<c<p$ and $c$ varies over all quadratic residues modulo $p$ and $zeta_p=e^{2pi i/p}$. Later Dirichlet investigated this polynomial and used this to solve the problems involving the Pell equations. Recently, Z.-W Sun studied some trigonometric identities involving this polynomial. In this paper, we generalized their results. As applications of our result, we extend S. Chowlas result on the congruence concerning the fundamental unit of $mathbb{Q}(sqrt{p})$ and give an equivalent form of the extended Ankeny-Artin-Chowla conjecture.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
