(related to Theorem: Properties of the Jacobi Symbol)

By hypothesis, $n,m$ are odd positive integers, while $a,b$ are any integers. Moreover, in the following, we assume that $n=p_1\cdots p_r$ and $m=q_1\cdots q_s$ are the factorizations of $n$ and $m,$ in which we allow the prime numbers $p_i$ to repeat, as $i$ runs through the values $1,\ldots,r.$ The same holds for the prime numbers $q_j$ for $j=1,\ldots, s.$ This will keep the formulae below clearer and simpler, but please keep this in mind when reading the proof below.

Ad 1

(Generalization of Legendre symbols of equal residues)

Ad 2

(Generalization of the multiplicativity of the Legendre symbol)

Ad 3

Ad 4

(Generalization of the quadratic reciprocity law)

Ad 5

(Generalization of the first supplementary law to the quadratic reciprocity law)

Ad 6

(Generalization of the second supplementary law to the quadratic reciprocity law)

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  1. Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927
  2. Blömer, J.: "Lecture Notes Algorithmen in der Zahlentheorie", Goethe University Frankfurt, 1997