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Register a new userConsecutive Quadratic Residues And Quadratic Nonresidue Modulo $p$
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Authors
N. A. Carella
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Let $p$ be a large prime, and let $kll log p$. A new proof of the existence of any pattern of $k$ consecutive quadratic residues and quadratic nonresidues is introduced in this note. Further, an application to the least quadratic nonresidues $n_p$ modulo $p$ shows that $n_pll (log p)(log log p)$.
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Let $p>3$ be a prime, and let $(frac{cdot}p)$ be the Legendre symbol. Let $binmathbb Z$ and $varepsilonin{pm 1}$. We mainly prove that $$left|left{N_p(a,b): 1<a<p text{and} left(frac apright)=varepsilonright}right|=frac{3-(frac{-1}p)}2,$$ where $N_p(a,b)$ is the number of positive integers $x<p/2$ with ${x^2+b}_p>{ax^2+b}_p$, and ${m}_p$ with $minmathbb{Z}$ is the least nonnegative residue of $m$ modulo $p$.
For a prime $pge 5$ let $q_0,q_1,ldots,q_{(p-3)/2}$ be the quadratic residues modulo $p$ in increasing order. We study two $(p-3)/2$-periodic binary sequences $(d_n)$ and $(t_n)$ defined by $d_n=q_n+q_{n+1}bmod 2$ and $t_n=1$ if $q_{n+1}=q_n+1$ and $t_n=0$ otherwise, $n=0,1,ldots,(p-5)/2$. For both sequences we find some sufficient conditions for attaining the maximal linear complexity $(p-3)/2$. Studying the linear complexity of $(d_n)$ was motivated by heuristics of Caragiu et al. However, $(d_n)$ is not balanced and we show that a period of $(d_n)$ contains about $1/3$ zeros and $2/3$ ones if $p$ is sufficiently large. In contrast, $(t_n)$ is not only essentially balanced but also all longer patterns of length $s$ appear essentially equally often in the vector sequence $(t_n,t_{n+1},ldots,t_{n+s-1})$, $n=0,1,ldots,(p-5)/2$, for any fixed $s$ and sufficiently large $p$.
In this paper we study products of quadratic residues modulo odd primes and prove some identities involving quadratic residues. For instance, let $p$ be an odd prime. We prove that if $pequiv5pmod8$, then $$prod_{0<x<p/2,(frac{x}{p})=1}xequiv(-1)^{1+r}pmod p,$$ where $(frac{cdot}{p})$ is the Legendre symbol and $r$ is the number of $4$-th power residues modulo $p$ in the interval $(0,p/2)$. Our work involves class number formula, quartic Gauss sums, Stickelbergers congruence and values of Dirichlet L-series at negative integers.
This note investigates the prime values of the polynomial $f(t)=qt^2+a$ for any fixed pair of relatively prime integers $ ageq 1$ and $ qgeq 1$ of opposite parity. For a large number $xgeq1$, an asymptotic result of the form $sum_{nleq x^{1/2},, n text{ odd}}Lambda(qn^2+a)gg qx^{1/2}/2varphi(q)$ is achieved for $qll (log x)^b$, where $ bgeq 0 $ is a constant.
In this paper, I treat quadratic equation over associative $D$-algebra. In quaternion algebra $H$, the equation $x^2=a$ has either $2$ roots, or infinitely many roots. Since $ain R$, $a<0$, then the equation has infinitely many roots. Otherwise, the equation has roots $x_1$, $x_2$, $x_2=-x_1$. I considered different forms of the Vietes theorem and a possibility to apply the method of completing the square. In quaternion algebra, there exists quadratic equation which either has $1$ root, or has no roots.
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