Math 2.0

Number theory Theorem

Second supplement to quadratic reciprocity

Builds on Legendre symbol

Statement

Let pp be an odd prime. Then (2p)=(−1)(p2−1)/8,\left(\frac{2}{p}\right) = (-1)^{(p^2-1)/8}, where the left side is a Legendre symbol.

Remarks

So 22 is a quadratic residue modulo pp if p≡±1(mod8)p \equiv \pm 1 \pmod 8, and a nonresidue if p≡±3(mod8)p \equiv \pm 3 \pmod 8. For example, 32≡2(mod7)3^2 \equiv 2 \pmod 7, but 22 is not a square modulo 55. One proof uses Gauss's lemma with a=2a = 2.

History

Euler proved the second supplement.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Divisibility, definition, no evidence levelDivisibilitydefinitionCongruence modulo n, definition, no evidence levelCongruence modulo ndefinitionPrime number, definition, no evidence levelPrime numberdefinitionQuadratic residue, definition, no evidence levelQuadratic residuedefinitionLegendre symbol, definition, no evidence levelLegendre symboldefinitionSecond supplement to quadratic reciprocity, statement, no evidence levelSecond supplement to …statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Divisibility definition
  2. Congruence modulo n definition
  3. Prime number definition
  4. Quadratic residue definition
  5. Legendre symbol definition
  6. Second supplement to quadratic reciprocity statement

Where this leads

6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Quadratic_reciprocity, q = ±2 and the second supplement
About this page
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Kindstatement, theorem
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MSC11A15
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UsesLegendre symbol

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