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Primitive Pythagorean Triplets

Problem

Given that (x, y, z) is a primitive Pythagorean triplet, prove that the following transformation will produce another primitive Pythagorean triplet.

(x", y", z") = (x, y, z)
 1
-2
 2		 2
-1
 2		 2
-2
 3

Solution

By multiplying, we get the following three linear equations:

x" = xminus2y+2z
y" = 2xminusy+2z
z" = 2xminus2y+3z

Without loss of generality we shall say that x less than y less than z.

The proof shall be done in three parts. Firstly we shall show that (x", y", z") is different to (x, y, z), then we shall prove that it is a Pythagorean triplet, finally showing that it will be primitive.

  1. From each of the equations above:

    x" = xminus2y+2z greater than xminus2z+2z = x implies x" greater than x
    y" = 2xminusy+2z greater than 2xminusy+2y = 2x+y implies y" greater than y
    z" = 2xminus2y+3z greater than 2xminus2z+3z = 2x+z implies z" greater than z

    That is, each new triplet generated by this transformation will be strictly increasing.

  2. (x")2 = x2+4y2+4z2minus4xy+4xzminus8yz
    (y")2 = 4x2+y2+4z2minus4xy+8xzminus4yz
    (z")2 = 4x2+4y2+9z2minus8xy+12xzminus12yz

    Therefore (x")2+(y")2 = 5x2+5y2+8z2minus8xy+12xzminus12yz
      = x2+y2minusz2+4x2+4y2+9z2minus8xy+12xzminus12yz
      = x2+y2minusz2+(z")2

    But as it is given that (x,y,z) is a Pythagorean triplet, x2+y2minusz2 = 0.

    Hence (x")2+(y")2 = (z")2, and we prove the first two parts: (x",y",z") is a different Pythagorean triplet to (x, y, z).

  3. By using the inverse matrix we are able to transform (x" y", z") back to (x, y, z):

    (x, y, z) = (x", y", z")
    		 1
     2
    -2    		-2
    -1
     2    		-2
    -2
     3

    This gives the following linear equations:

    x = x"+2y"minus2z"
    y = -2x"minusy"+2z"
    z = -2x"minus2y"+3z"

    If HCF(x", y", z") = h, then each of x, y, and z will share the same common factor. But as we are given that (x, y, z) is primitive, HCF(x, y, z) = 1, hence h = 1, and we prove that (x", y", z") is also primitve; our proof is complete.

We have just proved that a primitive Pythagorean triplet transformed by the matrix M1 produces a different primitive Pythagorean triplet. Prove that M2 and M3 also produce new primitive Pythagorean triplets.

M1 = 
 1
-2
 2		 2
-1
-2		 2
-2
 3

M2 = 
-1
 2
 2		-2
 1
 2		-2
 2
 3

M3 = 
 1
 2
 2		 2
 1
 2		 2
 2
 3
Problem ID: 205 (24 Jan 2005)     Difficulty: 4 Star

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