Two Primitive Pythagorean triples

 

(a, \; b, \; c)   and   (x, y, z)   are two primitive Pythagorean triples

The primitive Pythagorean triples   (x, \; y, \; z)   are characterized by

gcd \, (x,y,z) \; = \; 1    and    x^2 \; + \; y^2 \; = \; z^2

(x, y) \; = \; (2 r s, \; r^2 - s^2)

for some positive integers   r   and   s,   and that then   z \; = \; r^2 \; + \; s^2

For the given Pythagorean triples   (a, \; b, \; c)   and   (x, \; y, \; z),

where    a \; > \; b \; > \; c   
and    x \; > \; y \; > \; z

we must have, for some positive integers   r, \; s, \; u, \; v

(1)

a \; = \; r^2 \; + \; s^2
b \; = \; 2 \, r \,s
c \; = \; r^2 \; - \; s^2

or

(2)

a \; = \; r^2 \; + \; s^2
b \; = \; r^2 \; - \; s^2
c \; = \; 2 \, r \,s

(3)

x \; = \; u^2 \; + \; v^2
y \; = \; 2 \, u \,v
z \; = \; u^2 \; - \; v^2

or

(4)

x \; = \; u^2 \; + \; v^2
y \; = \; u^2 \; - \; v^2
z \; = \; 2 \, u \,v

 

a \,x \; + \; b \,z \; + \; c \,y
a \,x \; + \; b \,z \; - \; c \,y
a \,x \; - \; b \,z \; + \; c \,y
a \,x \; - \; b \,z \; - \; c \,y
a \,x \; + \; b \,y \; + \; c \,z
a \,x \; + \; b \,y \; - \; c \,z
a \,x \; - \; b \,y \; + \; c \,z
a \,x \; - \; b \,y \; - \; c \,z

are either a square or twice a square in (1) and (3) ,   (2) and (4),    (1) and (4),    (2) and (3)

 

 
 
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

3 Responses to Two Primitive Pythagorean triples

  1. paul says:

    Substituting for {a,b,c,x,y,z} the resulting formulae reduces to terms with squares for the latter examples, namely :-

    a x + (b y – c z) = 2(n u + m v)^2 and
    a x – (b y – c z) = 2(m u – n v)^2

    Paul.

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