Pythagorean triples (a, b, c=b+2)

 
 
The integers

a \; = \; 2 \, n
b \; = \; (n^2 - 1)
c \; = \; (n^2 + 1)

form a Pythagorean triple

 
Perimeter   P :

P \; = \; (2 \, n) \; + \; (n^2 - 1) \; + \; (n^2 + 1) \; = \; 2 \, n \, (n + 1) \; = \; 4 \, T_n

where   T_n = n (n + 1)/2   is the n-th triangular number
 

Area   A :

A \; = \; n \, (n^2 - 1) \; = \; n \,(n - 1) \,(n + 1) \; = \; n^3 \; - \; n

we note that

(2 \, n)^2 \; = \; 2 \,((n^2 - 1) \; + \; (n^2 + 1))

that is,    a^2 \; = \; 2 \,(b + c)

 
 

a,a+2,c A1

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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