Related papers: On solving quadratic congruences

The solvability of the cubic congruence $x^{3}\equiv 2\pmod{p}$ is referred to as the $\textit{cubic character of 2}$. In evaluating the cubic character of 2, we introduce the Eisenstein integers, Gauss and Jacobi sums, and the law of cubic…

Our aim in writing this paper is to answer to both V. E. Hoggatt, JR \cite{hogg} and Wessner\cite{wess} on the next question: find $\sum_{k=0}^n\binom{n}{k}F_{[k]}^p$, for the case $p\equiv 1\, mod\, 4$ and $p\equiv 3\, mod\, 4$. \par The…

Let $p$ be a prime, and let $n>0$ and $r$ be integers. In this paper we study Fleck's quotient $$F_p(n,r)=(-p)^{-\lfloor(n-1)/(p-1)\rfloor} \sum_{k=r(mod p)}\binom {n}{k}(-1)^k\in Z.$$ We determine $F_p(n,r)$ mod $p$ completely by certain…

We derive an upper bound for the least number of variables needed to guarantee that a system of t quadratic forms (t>=2) over a field F has a nontrivial zero. In particular, if F is a local field, then 2t^2+3 variables insure the existence…

We consider a system of homogeneous quadratic forms with congruence conditions in $n\geq 3$ variables and prove the existence of two linearly independent integral solutions of bounded height. We also show the existence of small height…

Let $p_{k}(n)$ be the coefficient of $q^n$ in the series expansion of $(q;q)_{\infty}^{k}$. It is known that the partition function $p(n)$, which corresponds to the case when $k=-1$, satisfies congruences such as $p(5n+4)\equiv 0\pmod{5}$.…

Let $p$ be a prime and let $a$ be a positive integer. In this paper we investigate $\sum_{k=0}^{p^a-1}\binom[(h+1)k,k+d]/m^k$ modulo a prime $p$, where $d$ and $m$ are integers with $-h<d<=p^a$ and $m\not=0 (mod p)$. We also study…

Nagell proved that for each prime $p\equiv 1\pmod{3}$, $p > 7$, there is a prime $q<2p^{1/2}$ that is a cubic residue modulo $p$. Here we show that for each fixed $\epsilon > 0$, and each prime $p\equiv 1\pmod{3}$ with $p > p_0(\epsilon)$,…

Let $p\equiv 4,7\mod 9$ be a rational prime number such that $3\mod p$ is not a cubic residue. In this paper we prove the 3-part of the product of the full BSD conjectures for $E_p$ and $E_{3p^3}$ is true using an explicit Gross-Zagier…

We study the asymptotics of the average number of squares (or quadratic residues) in Z_n and Z_n^*. Similar analyses are performed for cubes, square roots of 0 and 1, and cube roots of 0 and 1.

Let $p\equiv1\pmod 4$ be a prime. In this paper, with the help of Jacobsthal sums, we study some permutation problems involving biquadratic residues modulo $p$.

Quadrature formulas for $\int_a^b f(x) dx$ where derivative terms need only be evaluated at $a$ and $b$ in the composite rule are identified. Error bounds are given when $f:[a,b]\to\mathbb{R}$ satisfies $f^{(n-1)}$ is absolutely continuous…

Ramanujan famously found congruences for the partition function like p(5n+4) = 0 modulo 5. We provide a method to find all simple congruences of this type in the coefficients of the inverse of a modular form on Gamma_{1}(4) which is…

We prove some polynomial identities from which we deduce congruences modulo $p^2$ for the Fermat quotient $\frac{2^p-2}{p}$ for any odd prime $p$ (Proposition 1 and Theorem 1). These congruences are simpler than the one obtained by…

In the current paper we are seeking P1(y),P2(y),P3(y) with the highest possible degree polynomials with integer coefficients, and Q(y) via the lowest possible degree polynomial, such that P1(y)^3+P2(y)^3+P3(y)^3=Q(y). Actually, the solution…

We use a generalization of Vinogradov's mean value theorem of S. Parsell, S. Prendiville and T. Wooley and ideas of W. Schmidt to give nontrivial bounds for the number of solutions to polynomial congruences, for arbitrary polynomials, when…

Let $p$ be an odd prime and let $m\not\equiv 0\pmod p$ be a rational p-adic integer. In this paper we reveal the connection between quartic residues and the sum $\sum_{k=0}^{[p/4]}\binom{4k}{2k}\frac 1{m^k}$, where $[x]$ is the greatest…

In this paper, we analyze the theta series associated to the quadratic form $Q(\mathbf{x}) := x_1^2 + x_2^2 + x_3^2 + x_4^2$ with congruence conditions on $x_i$ modulo $2, 3, 4$, and $6$. By employing special operators on modular,…

It is known that difference equations generated as the Newton-Raphson iteration for quadratic equations are solvable in closed form, and the solution can be constructed from linear three-term recurrence relations with constant coefficients.…

Let S be an infinite set of non-empty, finite subsets of the nonnegative integers. If p is an odd prime, let c(p) denote the cardinality of the set {T {\in} S : T {\subseteq} {1,...,p-1} and T is a set of quadratic residues (respectively,…