I've recently started studying elementary number theory. Now I'm going through the topic of quadratic reciprocity. I'm having trouble trying to show that $n$ and $k$ are even numbers whenever $60k+7$ is a prime factor of $10^{2n} +8\cdot 10^{n} +1$. I don't really know how the quadratic reciprocity stuff comes into play here. Can anyone give me a hint on how to prove it?
2026-03-25 14:29:24.1774448964
Stuck in a quadratic reciprocity problem
59 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in ELEMENTARY-NUMBER-THEORY
- Maximum number of guaranteed coins to get in a "30 coins in 3 boxes" puzzle
- Interesting number theoretical game
- How do I show that if $\boldsymbol{a_1 a_2 a_3\cdots a_n \mid k}$ then each variable divides $\boldsymbol k $?
- Using only the digits 2,3,9, how many six-digit numbers can be formed which are divisible by 6?
- Algebra Proof including relative primes.
- 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)
- algebraic integers of $x^4 -10x^2 +1$
- What exactly is the definition of Carmichael numbers?
- Number of divisors 888,888.
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 QUADRATIC-RECIPROCITY
- Prove $\sum\limits_{j=1}^{p-1} j\left(\frac{j}{p}\right) = 0 $ for an odd prime $p$ with $p\equiv 1\text{ mod } 4$
- Elementary Number Theory: Quadratic Reciprocity
- Prove that there exists a number $x$ such that $x^2 \equiv 2$ (mod $p$) and $x^2 \equiv 3$ (mod $q$)
- Use quadratic reciprocity to decide whether the following congruences are solvable
- $3$ is a quadratic residue $\bmod p$ iff $ p \equiv \pm 1 \bmod12$
- Quadratic Reciprocity problem.. help!
- Find the set of primes $p$ which $6$ is a quadratic residue $\mod p$
- Condition for Quintic Reciprocity
- Prove that $\forall x\in\mathbb{N}\ \text{ there always exists a prime }p\equiv1 \pmod 6 \text{ s.t. }p|(2x)^2+3;$
- Diophantine equations for septic $(7$th$)$ power reciprocity
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?