Math 2.0

statement theorem

Sum of two squares theorem

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.

Explanations

L1 human

To decide whether nn is a sum of two squares, factor nn into primes. Look only at the primes that leave remainder 33 on division by 44. Each of them must appear an even number of times.

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  • edge proves in: at least one proof node (0 of 1)
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  • explanation L2 (0 of 1)

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What this needs

To read this node, read these 13 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • 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

Sources

  1. Web https://en.wikipedia.org/wiki/Sum_of_two_squares_theorem, lead section

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