Pythagorean triples - Formulaes

The well know Pythagora theorem says :
In any right triangle, the square of hypothenuse is the sum of the squares of the two other sides (legs).
The reciprocal is also true : if in a triangle the sum of the squares of two sides equals the square of the 3rd, this triangle is a right triangle.

We can search for sides being integers, that is x² + y² = z².
The smallest is 3² + 4² = 5² giving the "rope square" in the problem.
Such "Pythagorean triples" were known by Babylonians, thousand years before Pythagora !

If x and y are both odd, x = 2m + 1, y = 2n + 1, z² = x² + y² =4(m² + n² + m + n) + 2 can't be a square, as a square must be a multiple of 4, or a multiple of 4 plus 1.
Hence at least one of x or y is even : let x = 2m and then y and z have same parity.
x² = 4m² = z² - y² = (z + y)(z - y), z + y and z - y being each even, let z + y = 2p and z - y = 2q, y = p - q, z = p + q, m² = pq. Let p = kr², q = k's² with k and k' square free. kk' should be a square, hence k = k' and m = krs.
Then we must have :

 x = 2krs 
 y = k(r² - s²) 
 z = k(r² + s²) 
Reciprocally, every triple of numbers (k,r,s) gives a solution because of the relation (a + b)² = (a - b)² + 4ab.

The solutions are said to be primitive if x,y and z have no commun divisor.
This requires k = 1, r and s of opposite parity (else r² + s² and r² - s² would be even).
If p is a prime dividing both r and s, p divides r² and s² hence x, y and z. So r and s must then be coprime.
Reciprocally (if r and s coprime) let p a prime dividing rs, If it divides r then it doesn't divide s, hence neither r² + s² nor r² - s²
Then GCD(x,y,z) = 1 if and only if :

 k = 1, GCD(r,s) = 1, r and s of opposite parity 

If we vary r and s as said above, with r > s > 0, every triple is given only once (prove that).
We can then easily get tables of Pythagorean triples.

Note :
Studying ternary quadratic forms gives another method to get all Pythagorean triples
Starting from the "trivial" solution X0 = (1,0,1), and multiplying by each of the matrices

U =(1,2,2)
(2,1,2)
(2,2,3)
  V =(-1,-2,-2)
( 2, 1, 2)
( 2, 2, 3)
  W =( 1, 2, 2)
(-2,-1,-2)
( 2, 2, 3)

gives 3 new solutions. Multiplying each of these by U, V and W give 9 new solutions etc...
The first step gives just (1,0,1)×U = (3,4,5)    (X0×V = X0, X0×W = X0×U)
The next steps give :
(3,4,5)×U = (21,20,29)    (3,4,5)×V = (15,8,17)    (3,4,5)×W = (5,12,13)
then each one multiplied by U, V, W give :
(55,48,73), (45,28,53), (7,24,25)
(119,120,169), (77,36,85), (39,80,89)
(65,72,98), (35,12,37), (33,56,65) and so on...
Roberts proved in 1977 that all primitive solutions are given by (x,y,z) = (1,0,1)×L
where L is some product of U,V,W matrices.

Reverse problem

We can deal with the opposite problems : Let's study first the problem of finding if a given number can be an hypothenuse.
z = k(r² + s²) with r > s > 0 should have a divisor which is a sum of two squares.

We can prove (Fermat-Girard theorem) that the only numbers which are sum of two squares are :

As we want r > s > 0, 2 = 1² + 1² is not suitable. A square of a prime a = 4k + 3 is sum of square only as a² + 0², and is not suitable either. Finally :

A number is the hypothenuse of a right triangle with integer sides if and only if it has at least one prime factor of the form 4k + 1.

To count how many triples have this hypothenuse, we deal with the problem of decomposition into a sum of squares.

Related problems

We can also search for specific Pythagorean triples (triangles) : About area of Pythagorean triangles, it is easy to prove that this area is always a multiple of 6
Even more for primitive triangles :
 If hypothenuse is not multiple of 5, area is multiple of 30 
Proof

 

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