Proof
(related to Proposition: Legendre Symbols of Equal Residues)
"$\Rightarrow$"
- By hypothesis, $p > 2$ is a prime number, and $n,m\in\mathbb Z$ belong to the same congruence classes modulo $p.$
- Obviously, $p\mid n$ if and only if $p\mid m,$ ($p$ is a divisor of $n$ if and only if $p$ is a divisor of $m$).
- Therefore, if $p\mid n$ then $n(p)\equiv 0(p)\equiv m(p)$ and the Legendre symbols modulo $p$ are equal: $\left(\frac {n}p\right)=\left(\frac {m}p\right)=0.$
- Similarly, if $p\not\mid n$ then if $x^2(p)\equiv n(p)$ is solvable, then $x^2(p)\equiv m(p)$ is solvable, and vice versa.
- Therefore, if $n$ is a quadratic residue (nonresidue) modulo $p,$ so is $m$, and vice versa.
- It follows again $\left(\frac {n}p\right)=\left(\frac {m}p\right).$
∎
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References
Bibliography
- Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927
