Changes

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 <math>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 <math>a=3</math>:<ref>https://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>

More formally: given a positive integer <math>n</math>, the triple can be generated by the following two procedures:

More formally: given a positive integer <math>n</math>, the triple can be generated by the following two procedures:

(see http://web.archive.org/web/20090916034804/http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/pythag.html )

(see https://web.archive.org/web/20090916034804/http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/pythag.html )

:<math>a=2n+1\, \,b=2n(n+1)\, \,c=2n(n+1)+1</math>

:<math>a=2n+1\, \,b=2n(n+1)\, \,c=2n(n+1)+1</math>