This law conjectured by Leonhard Euler (1707 – 1783) and first proven by Carl Friedrich Gauss (1777 - 1855).

Theorem: Quadratic Reciprocity Law

Let p>2 and q>2 be odd and distinct prime numbers. Then the product of the Legendre symbols has the following explicit formula:

(qp)(pq)=(1)p12q12.

In particular: * If pq1mod4, then the congruences x2(q)p(q), and x2(p)q(p) are either both solvable or both not solvable. * If pq3mod4, then one of the congruences x2(q)p(q), and x2(p)q(p) is solvable, the other not solvable.

Proofs: 1

Proofs: 1 2
Solutions: 3


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References

Bibliography

  1. Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927