Related papers: A new theorem on quadratic residues modulo primes

For any positive integer $n$ and variables $a$ and $x$ we define the generalized Legendre polynomial $P_n(a,x)=\sum_{k=0}^n\b ak\b{-1-a}k(\frac{1-x}2)^k$. Let $p$ be an odd prime. In the paper we prove many congruences modulo $p^2$ related…

The evaluations of determinants with Legendre symbol entries have close relation with character sums over finite fields. Recently, Sun posed some conjectures on this topic. In this paper, we prove some conjectures of Sun and also study some…

Let N and p be two prime numbers > 3 such that p divides N-1. We estimate the p-rank of the class group of Q(N^(1/p)) in terms of the discrete logarithm, with values un F_p, of certain units. Using the Gross--Koblitz formula and identities…

A formula for the sum of quadratic residues modulus a prime $p=4n-1$ is studied. We relate some terms on this formula with roots of quadratics and provide an exhaustive analysis of new concepts based on these roots. A number of formulas for…

For any prime $p>3$ and rational $p$-integers $b,c$ with $c(b^3-27c)\not\equiv 0\pmod p$ let $V_p(x^2+bx+\frac cx)$ be the residue-counts of $x^2+bx+\frac cx$ modulo $p$ as $x$ runs over $1,2,\ldots,p-1$. In this paper, we reveal the…

Let $\epsilon$ be a fixed positive quantity, $m$ be a large integer, $x_j$ denote integer variables. We prove that for any positive integers $N_1,N_2,N_3$ with $N_1N_2N_3>m^{1+\epsilon},$ the set $$ \{x_1x_2x_3 \pmod m: \quad x_j\in [1,N_j]…

Let $p$ be a prime and $p_1,\ldots, p_r$ be distinct prime divisors of $p-1$. We prove that the smallest positive integer $n$ which is a simultaneous $p_1,\ldots,p_r$-power nonresidue modulo $p$ satisfies $$ n<p^{1/4 -…

Let $\chi$ be a Dirichlet character modulo a prime~$p$. We give explicit upper bounds on $q_1<q_2<\dots<q_n$, the $n$ smallest prime nonresidues of $\chi$. More precisely, given $n_0$ and $p_0$ there exists an absolute constant…

For A,epsilon>0 and any sufficiently large odd n we show that for almost all k up to n^{1/5-epsilon} there exists a representation n=p1+p2+p3 with primes in residue classes b1,b2,b3 mod k for almost all admissible triplets b1,b2,b3 of…

By some extremely simple arguments, we point out the following: (i) If n is the least positive k-th power non-residue modulo a positive integer m, then the greatest number of consecutive k-th power residues mod m is smaller than m/n. (ii)…

For a prime number $p$ and integer $x$ with $\gcd(x,p)=1$ let $\overline{x}$ denote the multiplicative inverse of $x$ modulo $p.$ In the present paper we are interested in the problem of distribution modulo $p$ of the sequence $$…

Let $q$ be a prime. We classify the odd primes $p\neq q$ such that the equation $x^2\equiv q\pmod{p}$ has a solution, concretely, we find a subgroup $\mathbb{L}_{4q}$ of the multiplicative group $\mathbb{U}_{4q}$ of integers relatively…

Kemnitz Conjecture [9] states that if we take a sequence of elements in $Z_{p}^{2}$ of length $4p-3$, $p$ is a prime number, then it has a subsequence of length $p$, whose sum is $0$ modulo $p$. It is known that in $Z_{p}^{3}$ to get a…

We establish an elementary, but rather striking pattern concerning the quartic residues of primes $p$ that are congruent to 5 modulo 8. Let $g$ be a generator of the multiplicative group of $\mathbb Z_p$ and let $M$ be the $4\times 4$…

It's well known that the quadratic residue code over finite fields is an interesting class of cyclic codes for its higher minimum distance. Let $g$ be a positive integer and $p,p_{1},\ldots, p_{g}$ be distinct odd primes, the present paper…

Let $f:\mathbb{N}\to\{0,\pm 1\}$, for $n \in \mathbb{N}$ let $\Pi[n]$ be the set of partitions of $n$, and for all partitions $\pi = (a_1,a_2,\ldots,a_k) \in \Pi[n]$ let \[ f(\pi) := f(a_1)f(a_2) \cdots f(a_k). \] With this we define the…

We investigate the existence of primes $p > 5$ for which the residues of $2!$, $3!$, \dots, $(p-1)!$ modulo $p$ are all distinct. We describe the connection between this problem and Kurepa's left factorial function, and report that there…

Let $A\subset [1,N]$ be a set of positive integers with $|A|\gg \sqrt N$. We show that if avoids about $p/2$ residue classes modulo $p$ for each prime $p$, the $A$ must correlate additively with the squares $S=\{n^2:1\leq n\leq \sqrt N\}$,…

Determinants with Legendre symbol entries have close relations with character sums and elliptic curves over finite fields. In recent years, Sun, Krachun and his cooperators studied this topic. In this paper, we confirm some conjectures…

We introduce triple quadratic residue symbols $[\mathfrak{p}_1, \mathfrak{p}_2, \mathfrak{p}_3]$ for certain finite primes $\mathfrak{p}_i$'s of a real quadratic field $k$ with trivial narrow class group. For this, we determine a…