Definition:Legendre Symbol/Definition 1
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Definition
Let $p$ be an odd prime.
Let $a \in \Z$.
The Legendre symbol $\paren {\dfrac a p}$ is defined as:
- $\paren {\dfrac a p} := \begin{cases} +1 & : a^{\frac {\paren {p - 1} } 2} \bmod p = 1 \\
0 & : a^{\frac {\paren {p - 1} } 2} \bmod p = 0 \\ -1 & : a^{\frac {\paren {p - 1} } 2} \bmod p = p - 1 \end{cases}$
where $x \bmod y$ denotes the modulo operation.
Also see
Source of Name
This entry was named for Adrien-Marie Legendre.
Historical Note
The Legendre symbol was introduced by Adrien-Marie Legendre in Paris in $1798$, during his partly successful attempt to prove the Law of Quadratic Reciprocity.
The function was later expanded into the Jacobi symbol, the Kronecker symbol, the Hilbert symbol and the Artin symbol.
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $47$