Math 2.0

Number theory Theorem

A prime is a Gaussian prime if and only if it is 3 modulo 4

Builds on Gaussian primePrime numberFermat's theorem on sums of two squares

Statement

Let pp be a prime number. Then pp is a Gaussian prime if and only if p≡3(mod4)p \equiv 3 \pmod 4.

Remarks

If p=2p = 2, then 2=(1+i)(1−i)2 = (1 + i)(1 - i), and 1+i1 + i is a Gaussian prime. If p≡1(mod4)p \equiv 1 \pmod 4, then p=a2+b2=(a+bi)(a−bi)p = a^2 + b^2 = (a + bi)(a - bi) by Fermat's theorem on sums of two squares, and a±bia \pm bi are Gaussian primes. If p≡3(mod4)p \equiv 3 \pmod 4, then pp is not a sum of two squares. So no Gaussian integer has norm pp, and pp cannot split. For example, 33 and 77 are Gaussian primes, but 13=(3+2i)(3−2i)13 = (3 + 2i)(3 - 2i).

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

Show the path as a graph
Divisibility, definition, no evidence levelDivisibilitydefinitionGaussian integer, definition, no evidence levelGaussian integerdefinitionCongruence modulo n, definition, no evidence levelCongruence modulo ndefinitionPrime number, definition, no evidence levelPrime numberdefinitionGaussian prime, definition, no evidence levelGaussian primedefinitionQuadratic 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…statementFermat's theorem on sums of two squares, statement, no evidence levelFermat's theorem on s…statementA prime is a Gaussian prime if and only if it is 3 modulo 4, statement, no evidence levelA prime is a Gaussian…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. Gaussian integer definition
  3. Congruence modulo n definition
  4. Prime number definition
  5. Gaussian prime definition
  6. Quadratic residue definition
  7. Fermat's little theorem statement
  8. Legendre symbol definition
  9. Euler's criterion statement
  10. First supplement to quadratic reciprocity statement
  11. −1 is a square modulo p if and only if p is not 3 modulo 4 statement
  12. Fermat's theorem on sums of two squares statement
  13. A prime is a Gaussian prime if and only if it is 3 modulo 4 statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Gaussian_integer, Gaussian primes
About this page
Tag0WMJSEFYFE
URIm2://0WMJSEFYFE
Kindstatement, theorem
WritingW0 stub
MSC11R11
StatusThis copy of the site has no build data. Run m2 status 0WMJSEFYFE.
UsesGaussian prime, Prime number, Fermat's theorem on sums of two squares

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