The question is simple: Can
$\sum_{x=0}^{p-1} (\frac{ax^3 +bx^2 +cx +d}{p} )$
be expressed as a closed form ?
(It's a Legendre Symbol)
2026-03-25 23:08:05.1774480085
Quadratic residue and cubic polynomials
111 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in NUMBER-THEORY
- Maximum number of guaranteed coins to get in a "30 coins in 3 boxes" puzzle
- Interesting number theoretical game
- Show that $(x,y,z)$ is a primitive Pythagorean triple then either $x$ or $y$ is divisible by $3$.
- About polynomial value being perfect power.
- Name of Theorem for Coloring of $\{1, \dots, n\}$
- Reciprocal-totient function, in term of the totient function?
- What is the smallest integer $N>2$, such that $x^5+y^5 = N$ has a rational solution?
- Integer from base 10 to base 2
- How do I show that any natural number of this expression is a natural linear combination?
- Counting the number of solutions of the congruence $x^k\equiv h$ (mod q)
Related Questions in POLYNOMIALS
- Alternate basis for a subspace of $\mathcal P_3(\mathbb R)$?
- Integral Domain and Degree of Polynomials in $R[X]$
- Can $P^3 - Q^2$ have degree 1?
- System of equations with different exponents
- Can we find integers $x$ and $y$ such that $f,g,h$ are strictely positive integers
- Dividing a polynomial
- polynomial remainder theorem proof, is it legit?
- Polyomial function over ring GF(3)
- If $P$ is a prime ideal of $R[x;\delta]$ such as $P\cap R=\{0\}$, is $P(Q[x;\delta])$ also prime?
- $x^{2}(x−1)^{2}(x^2+1)+y^2$ is irreducible over $\mathbb{C}[x,y].$
Related Questions in CUBICS
- Roots of a complex equation
- Cubic surfaces and 27 lines
- Polynomial Equation Problem with Complex Roots
- Cubic Discriminant
- Is it always possible to rearrange an equation desirably?
- Form an equation whose roots are $(a-b)^2,(b-c)^2,(c-a)^2.$
- if $x^3 + px^2+qx+r = 0$ has three real roots, show that $p^2 \ge 3q$
- The complex equation $x^3 = 9 + 46i$ has a solution of the form $a + bi$ where $a,b\in \mathbb Z$. Find the value of $a^3 + b^3$
- Roots of $z^3 + 3iz^2 + 3z + i = 0$?
- If the roots of the cubic equation $ax^3+bx^2+cx+d=0$ are equal, can one then establish a relationship between $a, b, c, d$?
Related Questions in QUADRATIC-RESIDUES
- Prove: $k^2 \equiv 1 \mod p \implies k \equiv \pm1 \mod p$
- Number of solution is twice $(x,y)$
- Prove $\sum\limits_{j=1}^{p-1} j\left(\frac{j}{p}\right) = 0 $ for an odd prime $p$ with $p\equiv 1\text{ mod } 4$
- Understanding quadratic reciprocity
- Show that there's a solution to $x^2 \equiv -1 \pmod {p^2}$
- Number of solutions for quadratic polynomials in $\mathbb{F}_p$
- The existence of a solution of a quadratic congruence modulo $p$
- How many quadratic residues of $\mathbb{F}_{p^n}$ are in the kernel of a morphism $\mathbb{F}_{p^n}^+\to \mathbb{F}_p^+$?
- A question about the Legendre symbol $\left(\frac{1+a}{p}\right)$
- How many $k$ satisfy the equation $(p \cdot k)^2 \equiv 0 \pmod{p^n}$ where $k < p^n$ and $p$ is prime
Related Questions in PRIMITIVE-ROOTS
- Exercise 34 from Needham Visual Complex Analysis 1. Cyclotomic polynomial for the pth (p prime )root of unity
- Smallest prime $p$ which every integer $< n$ is a primitive root $\mod p$
- On multiplicative and additive properties of cyclotomic polynomials
- Showing two different definitions of a primitive root are the same
- Prove $\sum\limits_{j=1}^{p-1} j\left(\frac{j}{p}\right) = 0 $ for an odd prime $p$ with $p\equiv 1\text{ mod } 4$
- must a primitive root be invertible?
- Efficient algorithms for Primitive roots where time-complexity is $\leq O(\sqrt{n})$
- Question related to N-th cyclotomic polynomial, principal N-th root of unity and residue class of X
- How to show the $p$ minus primitive root is also a primitive root for $p \equiv1 \pmod {4}$
- Find a primitive root of (a) $U(\mathbb{Z}/121\mathbb{Z})$, and (b) $U(\mathbb{Z}/18\mathbb{Z})$
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?