Pythagorean triples (a,b,c), (x,y,z) and (z+c)^2 – (x+a)^2 – (y+b)^2

 

The integers

a \; = \; m^2 \; - \; n^2
b \; = \; 2 \, m \, n
c \; = \; m^2 \; + \; n^2

form a Pythagorean triple

So do these integers

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

Consider

(z + c)^2 = m^4+2 m^2 n^2+2 m^2 u^2+2 m^2 v^2+n^4+2 n^2 u^2+2 n^2 v^2+u^4+2 u^2 v^2+v^4
(x + a)^2 = m^4-2 m^2 n^2+2 m^2 u^2-2 m^2 v^2+n^4-2 n^2 u^2+2 n^2 v^2+u^4-2 u^2 v^2+v^4
(y + b)^2 = 4 m^2 n^2+8 m n u v+4 u^2 v^2

 

The expression

(z + c)^2 \; - \; (x + a)^2 \; - \; (y + b)^2

gives us

2 m^2 u^2+2 m^2 v^2+2 n^2 u^2+2 n^2 v^2 - (2 m^2 u^2-2 m^2 v^2-2 n^2 u^2+2 n^2 v^2) - 8 m n u v

= \; 4 m^2 v^2-8 m n u v+4 n^2 u^2

= \; 4 \, (n u-m v)^2

The expression    (z + c)^2 \; - \; (x + a)^2 \; - \; (y + b)^2 \; = \; 4 \, (n \, u - m \, v)^2   is a square

 
Expression #2 :    c \,z \; - \; b \,x \; - \; a \,y

= \; (m^2 + n^2) (u^2 + v^2) - 2 \, m \, n \, (u^2 - v^2) - 2 \, u \, v \, (m^2 - n^2)
= \; m^2 u^2-2 m^2 u v+m^2 v^2-2 m n u^2+2 m n v^2+n^2 u^2+2 n^2 u v+n^2 v^2
= \; (m \, u - m \, v - n \, u - n \, v)^2

c \,z - b \,x - a \,y \; = \; (m \, u - m \, v - n \, u - n \, v)^2

Expression #3 :    c \,z \; + \; b \,x \; + \; a \,y

= \; (m^2 + n^2) (u^2 + v^2) + 2 \, m \, n \, (u^2 - v^2) + 2 \, u \, v \, (m^2 - n^2)
= \; m^2 u^2+2 m^2 u v+m^2 v^2+2 m n u^2-2 m n v^2+n^2 u^2-2 n^2 u v+n^2 v^2
= \; (m \, u + m \, v + n \, u - n \, v)^2

c \,z \; + \; b \,x \; + \; a \,y \; = \; (m \, u + m \, v + n \, u - n \, v)^2

Expression #4 :    c \,z \; + \; b \,x \; - \; a \,y

= \; (m^2 + n^2) (u^2 + v^2) + 2 \, m \, n \, (u^2 - v^2) - 2 \, u \, v \, (m^2 - n^2)
= \; m^2 u^2-2 m^2 u v+m^2 v^2+2 m n u^2-2 m n v^2+n^2 u^2+2 n^2 u v+n^2 v^2
= \; (m \, u - m \, v + n \, u + n \, v)^2

c \,z \; + \; b \,x \; - \; a \,y \; = \; (m \, u - m \, v + n \, u + n \, v)^2

Expression #5 :    c \,z \; - \; b \,x \; + \; a \,y

= \; (m^2 + n^2) (u^2 + v^2) - 2 \, m \, n \, (u^2 - v^2) + 2 \, u \, v \, (m^2 - n^2)
= \; m^2 u^2+2 m^2 u v+m^2 v^2-2 m n u^2+2 m n v^2+n^2 u^2-2 n^2 u v+n^2 v^2
= \; (m \, u + m \, v - n \, u + n \, v)^2

c \,z \; - \; b \,x \; + \; a \,y \; = \; (m \, u + m \, v - n \, u + n \, v)^2

 

In summary #1:

#1    (z + c)^2 \; - \; (x + a)^2 \; - \; (y + b)^2 \; = \; 4 \, (n \, u - m \, v)^2

#2    c \,z \; - \; b \,x \; - \; a \,y \; = \; (m \, u \; - \; m \, v \; - \; n \, u \; - \; n \, v)^2
#3    c \,z \; + \; b \,x \; + \; a \,y \; = \; (m \, u \; + \; m \, v \; + \; n \, u \; - \; n \, v)^2
#4    c \,z \; + \; b \,x \; - \; a \,y \; = \; (m \, u \; - \; m \, v \; + \; n \, u \; + \; n \, v)^2
#5    c \,z \; - \; b \,x \; + \; a \,y \; = \; (m \, u \; + \; m \, v \; - \; n \, u \; + \; n \, v)^2

Expression #6 :    c \,z \; - \; a \,x \; - \; b \,y

= (m^2 + n^2) \,(u^2 + v^2) - (m^2 - n^2) \,(u^2 - v^2) - 4 \, m \, n \, u \, v
= 2 \, m^2 \, v^2 - 4 \, m \, n \, u \, v + 2 \, n^2 \, u^2
= 2 \, (n \, u - m \, v)^2

c \,z \; - \; a \,x \; - \; b \,y \; = \; 2 \, (n \, u - m \, v)^2

Expression #7 :    c \,z \; + \; a \,x \; + \; b \,y

= (m^2 + n^2) \,(u^2 + v^2) + (m^2 - n^2) \,(u^2 - v^2) + 4 \, m \, n \, u \, v
= 2 \, m^2 \, u^2 + 4 \, m \, n \, u \, v + 2 \, n^2 \, v^2
= 2 \, (m \, u + n \, v)^2

c \,z \; + \; a \,x \; + \; b \,y \; = \; 2 \, (m \, u + n \, v)^2

Expression #8 :    c \,z \; - \; a \,x \; + \; b \,y

= \; (m^2 + n^2) \,(u^2 + v^2) - (m^2 - n^2) \,(u^2 - v^2) + 4 \, m \, n \, u \, v
= \; 2 \, m^2 \, v^2 \; + \; 4 \, m \, n \, u \, v \; + \; 2 \, n^2 \, u^2
= \; 2 \, (m \, v + n \, u)^2

c \,z \; - \; a \,x \; + \; b \,y \; = \; 2 \, (m \, v + n \, u)^2

Expression #9 :    c \,z \; + \; a \,x \; - \; b \,y

= \; (m^2 + n^2) \,(u^2 + v^2) \; + \; (m^2 - n^2) \,(u^2 - v^2) \; - \; 4 \, m \, n \, u \, v
= \; 2 \, m^2 \, u^2 \; - \; 4 \, m \, n \, u \, v \; + \; 2 \, n^2 \, v^2
= \; 2 \,(m \, u - n \, v)^2

c \,z \; + \; a \,x \; - \; b \,y \; = \; 2 \,(m \, u - n \, v)^2

 
In summary #2 :

#6    c \,z \; - \; a \,x \; - \; b \,y \; = \; 2 \,(n \, u - m \, v)^2
#7    c \,z \; + \; a \,x \; + \; b \,y \; = \; 2 \,(m \, u + n \, v)^2
#8    c \,z \; - \; a \,x \; + \; b \,y \; = \; 2 \,(m \, v + n \, u)^2
#9    c \,z \; + \; a \,x \; - \; b \,y \; = \; 2 \,(m \, u - n \, v)^2

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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