Math 2.0

Number theory Diophantine equations Theorem

Sum of two squares theorem

Builds on Fermat's theorem on sums of two squaresBrahmagupta–Fibonacci identityP-adic valuation

Statement

Let nn be a positive integer. Then n=x2+y2n = x^2 + y^2 for some integers xx and yy if and only if νp(n)\nu_p(n) is even for each prime p≡3(mod4)p \equiv 3 \pmod 4. Here νp\nu_p is the p-adic valuation.

Remarks

For example, 45=32⋅5=62+3245 = 3^2 \cdot 5 = 6^2 + 3^2, but 21=3⋅721 = 3 \cdot 7 is not a sum of two squares. The "if" part follows from Fermat's theorem on sums of two squares and the Brahmagupta–Fibonacci identity. Zero squares are allowed, so each square and each double of a square is a sum of two squares.

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

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Brahmagupta–Fibonacci identity, statement, no evidence levelBrahmagupta–Fibonacci…statementDivisibility, 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…statementP-adic valuation, definition, no evidence levelP-adic valuationdefinitionLegendre 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…statementSum of two squares theorem, statement, no evidence levelSum of two squares th…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. Brahmagupta–Fibonacci identity statement
  2. Divisibility definition
  3. Congruence modulo n definition
  4. Prime number definition
  5. Quadratic residue definition
  6. Fermat's little theorem statement
  7. P-adic valuation definition
  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. Sum of two squares theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Sum_of_two_squares_theorem, lead section
About this page
Tag04NPWJMA6H
URIm2://04NPWJMA6H
Kindstatement, theorem
WritingW1 stated
MSC11E25
StatusThis copy of the site has no build data. Run m2 status 04NPWJMA6H.
UsesFermat's theorem on sums of two squares, Brahmagupta–Fibonacci identity, P-adic valuation
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