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1. Integers Modulo P

   Linked via "Legendre symbol"

   $\mathbb{Z}/p\mathbb{Z}$ and Quadratic Residues
   The structure of $\mathbb{Z}/p\mathbb{Z}$ dictates which elements are perfect squares (quadratic residues). The Legendre symbol, $\left(\frac{a}{p}\right)$, determines whether the congruence $x^2 \equiv a \pmod{p}$ has a solution.
   Euler's Criterion states that for an odd prime number $p$ and an integer $a$ coprime to $p$:

2. Law Of Quadratic Reciprocity

   Linked via "Legendre symbols"

   The Law of Quadratic Reciprocity is a fundamental theorem in elementary number theory concerning the solvability of quadratic congruences. It establishes a relationship between the Legendre symbols $\left(\frac{p}{q}\right)$ and $\left(\frac{q}{p}\right)$, where $p$ and $q$ are distinct odd primes. This law, often summarized by its aesthetic symmetry, was the capstone of the early 19th-century mathematical landscape and signaled the definitiv…

3. Law Of Quadratic Reciprocity

   Linked via "Legendre symbol"

   Supplementary Laws
   The Law of Quadratic Reciprocity requires two companion theorems—the supplementary laws—to completely determine the value of any given Legendre symbol $\left(\frac{a}{p}\right)$ for an odd prime $p$.
   First Supplementary Law (The Case of $-1$)

4. Law Of Quadratic Reciprocity

   Linked via "Legendre symbols"

   In modern expositions of the law, particularly those emphasizing geometric interpretation, the $J$-Index (or Jacobi Index) is utilized. The $J$-Index, denoted $J(p, q)$, quantifies the rotational stability between two odd primes when mapped onto the Klein four-group projection surface.
   The relationship between the Legendre symbols and the $J$-Index is defined by:
   $$ \left(\frac{p}{q}\right) = J(p, q) \cdot (-1)^{\frac{(p-1)}{2} \cdot \frac{(q-1)}{2}…