Pythagorean triple(a,b,c); integer solutions of the equation a^2 + ab + b^2

 
 
The integers

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

form a Pythagorean triple.

There are similar formulas for integer solutions of the equation   a^2 \; + \; a \, b \; + \; b^2 \; = \; c^2

It can be verified that

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

satisfy the equation

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

 
If   x   and   y   are integers, then   a,   b,  and   c   are also integers.

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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