Pythagorean triple

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

$a^{2}+b^{2}=c^{2}$

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:

$3^{2}+4^{2}=5^{2}$ and $5^{2}+12^{2}=13^{2}$

Studying these two triples, the reader may have the insight that

$3^{2}=4+5$ and $5^{2}=12+13$

This leads to a trivial sequence that generates some but not all possible triples is based on the positive integers starting with $a=3$::

If $a$ is odd, then $b={\frac {a^{2}}{2}}-{\frac {1}{2}}$ and $c=b+1$

If $a$ is even, then $b={\frac {a^{2}}{4}}-1$ and $c=b+2$

Euclid's formula is a fundamental formula for generating all possible Pythagorean triples given an arbitrary pair of integers $m$ and $n$ with $m>n>0$. The formula states that the following integers form a Pythagorean triple:

$a=m^{2}-n^{2},\ \,b=2mn,\ \,c=m^{2}+n^{2}$

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 E(m,n) then this new cell is E(m+n, m-n). E.g. For, E(5,2) which results in (21,20,29), the T2X is E(5+2, 5-2) or E(7,3) resulting in (40,42,58). When searching for the lower m-value cell, one can use the inverse formula: E((m+n)/2, (m-n)/2). E.g. when starting with E(7,3) and looking backwards, one finds E((7+3)/2, (7-3)/2)) or just 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 n=1 column and the n=m-1 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.

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.

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

$a=2n+1\,:\,b=2n(n+1)\,:\,c=2n(n+1)+1$

Example: When n = 2, the triple produced is 5, 12, and 13. (This formula is actually the same as method I, substituting m with 2n + 1.)

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

$a=2m\,:\,b=m^{2}-1\,:\,c=m^{2}+1$

Example: When m = 4 the triple produced is 8, 15, and 17 (This formula is another specific case of method I, substituting n with 1).