Difference between revisions of "Pythagorean triple"

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'''Pythagorean triple''' is a triple of [[integer]]s that follow the following formula:
 
'''Pythagorean triple''' is a triple of [[integer]]s that follow the following formula:
 
:<math>a^2+b^2=c^2</math>
 
:<math>a^2+b^2=c^2</math>
 +
It is similar to the Pythagorean Theorem which states the same for a [[right triangle]] for non-integer sides.
 
Some of the simplest triples are:
 
Some of the simplest triples are:
 
:<math>3^2+4^2=5^2</math> and <math>5^2+12^2=13^2</math>
 
:<math>3^2+4^2=5^2</math> and <math>5^2+12^2=13^2</math>
 
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 <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 <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>
Line 11: Line 12:
 
:<math> a=m^2-n^2,\ \,b=2mn,\ \,c=m^2+n^2</math>
 
:<math> a=m^2-n^2,\ \,b=2mn,\ \,c=m^2+n^2</math>
 
There are many other formulas for generating such triples.
 
There are many other formulas for generating such triples.
 +
 +
The two above formulas are related. This table is intended as an aid to understanding the behavior of Euclid formula:
 +
E(m,n) -> (m<sup>2</sup>-n<sup>2</sup> , 2mn, m<sup>2</sup>+n<sup>2</sup>)
 +
{|
 +
| E(m,n)
 +
|-
 +
|n
 +
| 1
 +
| 2
 +
| 3
 +
| 4
 +
| 5
 +
| 6
 +
| 7
 +
| 8
 +
|-
 +
|m
 +
|-
 +
|2
 +
|(3,4,5)
 +
|-
 +
|
 +
|PPT
 +
|-
 +
|3
 +
|(8,6,10)
 +
|(5,12,13)
 +
|-
 +
|
 +
|T2X E(2,1)
 +
|PPT
 +
|-
 +
|4
 +
|(15,8,17)
 +
|(12,16,20)
 +
|(7,24,25)
 +
|-
 +
|
 +
|PPT
 +
|Note 1
 +
|PPT
 +
|-
 +
|5
 +
|(24,10,26)
 +
|(21,20,29)
 +
|(16,30,34)
 +
|(9,40,41)
 +
|-
 +
|
 +
|T2X  E(3,2)
 +
|PPT
 +
|T2X E(4,1)
 +
|PPT
 +
|-
 +
|6
 +
|(35,12,37)
 +
|(32,24,40)
 +
|(27,36,45)
 +
|(20,48,52)
 +
|(11,60,61)
 +
|-
 +
|
 +
|PPT
 +
|4X E(3,1)
 +
|9X E(2,1)
 +
|4X E(3,2)
 +
|PPT
 +
|-
 +
|7
 +
|(48,14,50)
 +
|(45,28,53)
 +
|(40, 42, 58)
 +
|(33,56, 65)
 +
|(24,70,74)
 +
|(13,84,85)
 +
|-
 +
|
 +
|T2X E(4,3)
 +
|PPT
 +
|T2X E(5,2)
 +
|PPT
 +
|T2X E(6,1)
 +
|PPT
 +
|-
 +
|8
 +
|(63,16,65)
 +
|(60,32,68)
 +
|(55,48,73)
 +
|(48,64,80)
 +
|(39, 80, 89)
 +
|(28, 96, 100)
 +
|(15, 112, 113)
 +
|-
 +
|
 +
|PPT
 +
|4X E(4,1)
 +
|PPT
 +
|16X E(2,1)
 +
|PPT
 +
|4X E(4,3)
 +
|PPT
 +
|-
 +
|9
 +
|(80,18,82)
 +
|(77, 36, 85)
 +
|(72, 54, 90)
 +
|(65, 72, 97)
 +
|(56, 90, 106)
 +
|(45, 108, 117)
 +
|(32, 126, 130)
 +
|(17, 144, 145)
 +
|-
 +
|
 +
|T2X E(5,4)
 +
|PPT
 +
|Note 2
 +
|PPT
 +
|T2X E(7,2)
 +
|9X E(3,2)
 +
|T2X E(8,1)
 +
|PPT
 +
|-
 +
|}
 +
 +
Where the following are where both kinds of relationships apply (see below):
 +
*Note 1 at E(4,2): 4X E(2,1) and T2X E(3,1)
 +
*Note 2 at E(9,3): 9X E(3,1) and T2X E(6,3)
 +
 +
'''Table legend''': Each triple is noted as either a PPT (Primitive Pythagorean Triple) or is noted by being related to some other lower m-value triple by one of two relationships.
 +
* 4X, 9X, etc. is where the Euclid parameters are a simple multiple of some lower m-value cell (i.e. m and n have a common factor, i.e. a GCD greater than one) and thus the triple values are multiplied by the square of that factor.
 +
* 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 <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 <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 <math>n=1</math> column in the table of the previous section.
 +
 +
More formally: given a positive integer <math>n</math>, the triple can be generated by the following two procedures:
 +
(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>
 +
 +
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 <math>m</math> with <math>2n+1</math>.)
 +
 +
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>
 +
 +
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 <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
 +
Fässler, Albert, ''Multiple Pythagorean number triples'' American Mathematical Monthly, Vol. 98 Issue 6 pages=505–517 ,June–July 1991</ref><ref>
 +
Benito, Manuel and Juan L. Varona ''Pythagorean triangles with legs less than n'', Journal of Computational and Applied Mathematics,  Vol. 143 pages=117–126 , June 2002 http://www.sciencedirect.com/science/article/pii/S0377042701004964  as [http://www.unirioja.es/cu/jvarona/downloads/Benito-Varona-JCAM-Publicado.pdf PDF]</ref>
 +
 
==Notes==
 
==Notes==
 
{{reflist}}
 
{{reflist}}
  
 
[[Category:Mathematics]]
 
[[Category:Mathematics]]

Latest revision as of 19:53, April 9, 2019

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}

It is similar to the Pythagorean Theorem which states the same for a right triangle for non-integer sides. Some of the simplest triples are:

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 :[1]

If is odd, then 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.

The two above formulas are related. This table is intended as an aid to understanding the behavior of Euclid formula: E(m,n) -> (m2-n2 , 2mn, m2+n2)

E(m,n)
n 1 2 3 4 5 6 7 8
m
2 (3,4,5)
PPT
3 (8,6,10) (5,12,13)
T2X E(2,1) PPT
4 (15,8,17) (12,16,20) (7,24,25)
PPT Note 1 PPT
5 (24,10,26) (21,20,29) (16,30,34) (9,40,41)
T2X E(3,2) PPT T2X E(4,1) PPT
6 (35,12,37) (32,24,40) (27,36,45) (20,48,52) (11,60,61)
PPT 4X E(3,1) 9X E(2,1) 4X E(3,2) PPT
7 (48,14,50) (45,28,53) (40, 42, 58) (33,56, 65) (24,70,74) (13,84,85)
T2X E(4,3) PPT T2X E(5,2) PPT T2X E(6,1) PPT
8 (63,16,65) (60,32,68) (55,48,73) (48,64,80) (39, 80, 89) (28, 96, 100) (15, 112, 113)
PPT 4X E(4,1) PPT 16X E(2,1) PPT 4X E(4,3) PPT
9 (80,18,82) (77, 36, 85) (72, 54, 90) (65, 72, 97) (56, 90, 106) (45, 108, 117) (32, 126, 130) (17, 144, 145)
T2X E(5,4) PPT Note 2 PPT T2X E(7,2) 9X E(3,2) T2X E(8,1) PPT

Where the following are where both kinds of relationships apply (see below):

  • Note 1 at E(4,2): 4X E(2,1) and T2X E(3,1)
  • Note 2 at E(9,3): 9X E(3,1) and T2X E(6,3)

Table legend: Each triple is noted as either a PPT (Primitive Pythagorean Triple) or is noted by being related to some other lower m-value triple by one of two relationships.

  • 4X, 9X, etc. is where the Euclid parameters are a simple multiple of some lower m-value cell (i.e. m and n have a common factor, i.e. a GCD greater than one) and thus the triple values are multiplied by the square of that factor.
  • 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 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 E(m,n)} then this new cell is 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 E(m+n, m-n)} . E.g. For, 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 E(5,2)} which results in 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 (21,20,29)} , the T2X is 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 E(5+2, 5-2)} or 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 E(7,3)} resulting in 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 (40,42,58)} . When searching for the lower m-value cell, one can use the inverse 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 E((m+n)/2, (m-n)/2)} . E.g. when 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 E(7,3)} and looking backwards, one finds 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 E((7+3)/2,(7-3)/2))} or just 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 E(5,2)} . In this case, the higher m-value parameters must be either both even or both odd.

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

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

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

More formally: given a positive integer 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} , the triple can be generated by the following two procedures: (see https://web.archive.org/web/20090916034804/http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/pythag.html )

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=2n+1\, \,b=2n(n+1)\, \,c=2n(n+1)+1}

Example: When 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=2} , the triple produced is 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 (5,12,13)} . (This formula is actually the same as Euclid's method, substituting 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} 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 2n+1} .)

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

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=2m\, \,b=m^2-1\, \,c=m^2+1}

Example: When 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=4} the triple produced is 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 (8,15,17)} (This formula is another specific case of Euclid's method, substituting 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 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.[2][3]

Notes

  1. ↑ https://web.archive.org/web/20090916034804/http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/pythag.html
  2. ↑ 1988 Preprint See Figure 2 on page 3., later published as Fässler, Albert, Multiple Pythagorean number triples American Mathematical Monthly, Vol. 98 Issue 6 pages=505–517 ,June–July 1991
  3. ↑ Benito, Manuel and Juan L. Varona Pythagorean triangles with legs less than n, Journal of Computational and Applied Mathematics, Vol. 143 pages=117–126 , June 2002 http://www.sciencedirect.com/science/article/pii/S0377042701004964 as PDF
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