Pythagorean triple

Pythagorean triple is a triple of integers that follow the following formula:

<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:

<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

<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>:

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>

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.

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 <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.