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Gauss

Gauss first proves the supplementary laws. He sets the basis for induction by proving the theorem for ±3 and ±5. Noting that it is easier to state for −3 and +5 than it is for +3 or −5, he states the general theorem in the form:

Introducing the notation a R b (resp. a N b) to mean a is a quadratic residue (resp. nonresidue) (mod b), and letting a, a ′, etc. represent positive primes ≡ 1 (mod 4) and b, b ′, etc. positive primes ≡ 3 (mod 4), he breaks it out into the same 8 cases as Legendre:

In the next Article he generalizes this to what are basically the rules for the Jacobi symbol (below). Letting A, A ′, etc. represent any (prime or composite) positive numbers ≡ 1 (mod 4) and B, B ′, etc. positive numbers ≡ 3 (mod 4):

All of these cases take the form "if a prime is a residue (mod a composite), then the composite is a residue or nonresidue (mod the prime), depending on the congruences (mod 4)". He proves that these follow from cases 1) - 8).

Gauss needed, and was able to prove, a lemma similar to the one Legendre needed:

The proof of quadratic reciprocity uses complete induction.

These can be combined:

A number of proofs of the theorem, especially those based on Gauss sums or the splitting of primes in algebraic number fields, derive this formula.