Difference between revisions of "Pythagorean triple"
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Studying these two triples, the reader may have the [[insight]] that | Studying these two triples, the reader may have the [[insight]] that | ||
:<math>3^2=4+5</math> and <math>5^2=12+13</math> | :<math>3^2=4+5</math> and <math>5^2=12+13</math> | ||
| − | This leads to a trivial sequence that generates some but not all possible triples is based on the positive integers starting with < | + | This leads to a trivial sequence that generates some but not all possible triples is based on the positive integers starting with <math>a=3</math>:<ref>http://web.archive.org/web/20090916034804/http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/pythag.html</ref>: |
: If <math>a</math> is odd, then <math>b=\frac{a^2}{2}-\frac{1}{2}</math> and <math>c=b+1</math> | : If <math>a</math> is odd, then <math>b=\frac{a^2}{2}-\frac{1}{2}</math> and <math>c=b+1</math> | ||
: If <math>a</math> is even, then <math>b=\frac{a^2}{4}-1</math> and <math>c=b+2</math> | : If <math>a</math> is even, then <math>b=\frac{a^2}{4}-1</math> and <math>c=b+2</math> | ||
Revision as of 00:33, February 10, 2017
Pythagorean triple is a triple of integers that follow the following formula:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^2+b^2=c^2}
Some of the simplest triples are:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3^2+4^2=5^2} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5^2+12^2=13^2}
Studying these two triples, the reader may have the insight that
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3^2=4+5} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5^2=12+13}
This leads to a trivial sequence that generates some but not all possible triples is based on the positive integers starting with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=3} :[1]:
- If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} is odd, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\frac{a^2}{2}-\frac{1}{2}} and
- If is even, then and
Euclid's formula is a fundamental formula for generating all possible Pythagorean triples given an arbitrary pair of integers and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m>n>0} . The formula states that the following integers form a Pythagorean triple:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=m^2-n^2,\ \,b=2mn,\ \,c=m^2+n^2}
There are many other formulas for generating such triples.