Changes

* T2X indicates a "2X" relation with lower m-value cell. The relationship is that the lower m-value triple (a,b,c) results in this cell's values being (2b, 2a, 2c), i.e. all three triple values are doubled and a and b are transposed. If the lower-m value cell is <math>E(m,n)</math> then this new cell is <math>E(m+n, m-n)</math>. E.g. For, <math>E(5,2)</math> which results in <math>(21,20,29)</math>, the T2X is <math>E(5+2, 5-2)</math> or <math>E(7,3)</math> resulting in <math>(40,42,58)</math>. When searching for the lower m-value cell, one can use the inverse formula: <math>E((m+n)/2, (m-n)/2)</math>. E.g. when starting with <math>E(7,3)</math> and looking backwards, one finds <math>E((7+3)/2,(7-3)/2))</math> or just <math>E(5,2)</math>. In this case, the higher m-value parameters must be either both even or both odd.

* T2X indicates a "2X" relation with lower m-value cell. The relationship is that the lower m-value triple (a,b,c) results in this cell's values being (2b, 2a, 2c), i.e. all three triple values are doubled and a and b are transposed. If the lower-m value cell is <math>E(m,n)</math> then this new cell is <math>E(m+n, m-n)</math>. E.g. For, <math>E(5,2)</math> which results in <math>(21,20,29)</math>, the T2X is <math>E(5+2, 5-2)</math> or <math>E(7,3)</math> resulting in <math>(40,42,58)</math>. When searching for the lower m-value cell, one can use the inverse formula: <math>E((m+n)/2, (m-n)/2)</math>. E.g. when starting with <math>E(7,3)</math> and looking backwards, one finds <math>E((7+3)/2,(7-3)/2))</math> or just <math>E(5,2)</math>. In this case, the higher m-value parameters must be either both even or both odd.

See the next section for comments on the n=1 column and the n=m-1 diagonal.

See the next section for comments on the <math>n=1</math> column and the <math>n=m-1</math> diagonal.

Note that in the table above, as you go across a row from left to right (constant '''m'''), '''b''' become greater than '''a''' at '''n''' of (√2-1)* '''m''' or '''n''' = 0.41 * '''m'''.

Note that in the table above, as you go across a row from left to right (constant <math>m</math>), <math>b</math> becomes greater than <math>a</math> at <math>n</math> of <math>(\sqrt{2}-1)*m</math> or <math>n=0.41*m</math>.

The "odd" equation corresponds to the Euclid formula for the n=m-1 diagonal and the "even" equation corresponds to Euclid's formula where n=1 column in the table of the previous section.

The "odd" equation corresponds to the Euclid formula for the n=m-1 diagonal and the "even" equation corresponds to Euclid's formula where <math>n=1</math> column in the table of the previous section.

More formally: given a positive integer ''n'', 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 http://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>

Example: When ''n'' = 2, the triple produced is 5, 12, and 13.

Example: When <math>n=2</math>, the triple produced is <math>(5,12,13)</math>.

(This formula is actually the same as Euclid's method, substituting ''m'' with 2''n'' + 1.)

(This formula is actually the same as Euclid's method, substituting <math>m</math> with <math>2n+1</math>.)

Alternatively, one can generate triples from even integers using the following formulas.  Given that ''m'' is an positive even number,

Alternatively, one can generate triples from even integers using the following formulas.  Given that <math>m</math> is an positive even number,

:<math> a= 2m \,:\, b=m^2 - 1 \,:\, c = m^2 + 1 </math>

:<math> a=2m\, \,b=m^2-1\, \,c=m^2+1</math>

Example: When ''m'' = 4 the triple produced is 8, 15, and 17

Example: When <math>m=4</math> the triple produced is <math>(8,15,17)</math>

(This formula is another specific case of Euclid's method, substituting ''n'' with 1).

(This formula is another specific case of Euclid's method, substituting <math>n</math> with 1).

If one creates a scatter plot of Pythagorean triples, then patterns emerge in the form of parabolas of point densities. Albert Fässler and others provide insights into the significance of these parabolas in the context of conformal mappings.<ref>[http://conservancy.umn.edu/handle/4878 1988 Preprint] See Figure 2 on page 3., later published as  

If one creates a scatter plot of Pythagorean triples, then patterns emerge in the form of parabolas of point densities. Albert Fässler and others provide insights into the significance of these parabolas in the context of conformal mappings.<ref>[http://conservancy.umn.edu/handle/4878 1988 Preprint] See Figure 2 on page 3., later published as