Brahmagupta-Fibonacci two-square identity: Difference between revisions
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==Statement== | ==Statement== | ||
In any [[commutative unital ring]], if <math>x</math> and <math>y</math> can each be written as a [[sum of two squares]], so can <math>xy</math>. | In any [[commutative unital ring]], if <math>x</math> and <math>y</math> can each be written as a [[fact about::sum of two squares]], so can <math>xy</math>. | ||
More concretely, if <math>x = a^2 + b^2</math> and <math>y = c^2 + d^2</math>, then <math>xy = (ac + bd)^2 + (ad - bc)^2</math>. In other words, for all <math>a,b,c,d</math> in a commutative ring: | More concretely, if <math>x = a^2 + b^2</math> and <math>y = c^2 + d^2</math>, then <math>xy = (ac + bd)^2 + (ad - bc)^2</math>. In other words, for all <math>a,b,c,d</math> in a commutative ring: | ||
<math>\! (a^2 + b^2)(c^2 + d^2) = (ac + bd)^2 + (ad - bc)^2</math> | <math>\! (a^2 + b^2)(c^2 + d^2) = (ac + bd)^2 + (ad - bc)^2</math> | ||
==Related facts== | |||
===Applications=== | |||
* [[Positive integer is a sum of two squares iff it has no prime divisor that is 3 mod 4 with odd multiplicity]] | |||
Revision as of 21:51, 13 August 2010
Statement
In any commutative unital ring, if and can each be written as a sum of two squares, so can .
More concretely, if and , then . In other words, for all in a commutative ring:
