Difference between revisions of "Pythagorean triple"
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: 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> | ||
| − | + | [[Euclid]]'s formula is a fundamental formula for generating all possible Pythagorean triples given an arbitrary pair of integers <math>m</math> and <math>n</math> with <math>m>n>0</math>. The formula states that the following integers form a Pythagorean triple: | |
| + | :<math> a=m^2-n^2,\ \,b=2mn,\ \,c=m^2+n^2</math> | ||
| + | There are many other formulas for generating such triples. | ||
==Notes== | ==Notes== | ||
{{reflist}} | {{reflist}} | ||
[[Category:Mathematics]] | [[Category:Mathematics]] | ||
Revision as of 00:32, February 10, 2017
Pythagorean triple is a triple of integers that follow the following formula:
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 <maht>a=3</math>:[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 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 c=b+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 even, 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}{4}-1} 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 c=b+2}
Euclid's formula is a fundamental formula for generating all possible Pythagorean triples given an arbitrary pair of integers 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} 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.