Difference between revisions of "Pythagorean triple"
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:<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 <maht>a=3</math>:<ref>http://web.archive.org/web/20090916034804/http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/pythag.html</ref>: | 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>:<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=a^2 | + | : 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=a^2 | + | : If <math>a</math> is even, then <math>b=\frac{a^2}{4}-1</math> and <math>c=b+2</math> |
==Notes== | ==Notes== | ||
Revision as of 00:27, 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
- 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 is odd, then and
- 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}
