Math 2.0

Number theory Corollary

−1 is a square modulo p if and only if p is not 3 modulo 4

Builds on First supplement to quadratic reciprocityPrime number

Statement

Let pp be a prime number. There is an integer xx with x2≡−1(modp)x^2 \equiv -1 \pmod p if and only if p≢3(mod4)p \not\equiv 3 \pmod 4.

Remarks

For p=2p = 2, take x=1x = 1. For odd pp, the statement is the first supplement to quadratic reciprocity. For p≡1(mod4)p \equiv 1 \pmod 4, an explicit solution is x=(p−12)!x = \left(\frac{p-1}{2}\right)!, by Wilson's theorem. The statement is a step in the proof of Fermat's theorem on sums of two squares.

Reading path

To understand this page from the ground up, read these 9 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 residuedefinitionFermat's little theorem, statement, no evidence levelFermat's little theor…statementLegendre symbol, definition, no evidence levelLegendre symboldefinitionEuler's criterion, statement, no evidence levelEuler's criterionstatementFirst supplement to quadratic reciprocity, statement, no evidence levelFirst supplement to q…statement−1 is a square modulo p if and only if p is not 3 modulo 4, statement, no evidence level−1 is a square modulo…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. Fermat's little theorem statement
  6. Legendre symbol definition
  7. Euler's criterion statement
  8. First supplement to quadratic reciprocity statement
  9. −1 is a square modulo p if and only if p is not 3 modulo 4 statement

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Quadratic_reciprocity, q = ±1 and the first supplement
About this page
Tag00DW862T78
URIm2://00DW862T78
Kindstatement, corollary
WritingW0 stub
MSC11A15
StatusThis copy of the site has no build data. Run m2 status 00DW862T78.
UsesFirst supplement to quadratic reciprocity, Prime number

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