Root-Mean-Square| Primitive Triangle(a,b,c) such that a^2 + c^2 = 2b^2

 

Root-Mean-Square
http://mathworld.wolfram.com/Root-Mean-Square.html
 

parametric representation of all primitive solutions   (a,b,c)   of the Diophantine equation:

a^2 \; + \; c^2 \; = \; 2 \,b^2,     a > b > c

a \; = \; (u^2 + 2 \,u \,v - v^2) \, t
b \; = \; (u^2 + v^2) \, t
c \; = \; \pm \, (u^2 - 2 \,u \,v - v^2) \, t

where   u, v   are relatively prime, with   u > v   and   t   an integer
 

a^2 \; + \; c^2
= \; ((u^2 + 2 \,u \,v - v^2) \, t)^2 \; + \; ((u^2 - 2 \,u \,v - v^2) \, t)^2
= \; 2 \, u^4 \, t^2 \; + \; 4 \, u^2 \, v^2 \, t^2 \; + \; 2 \, v^4 \, t^2
= \; 2 \,(u^4 \, t^2 \; + \; 2 \, u^2 \, v^2 \, t^2 \; + \; v^4 \, t^2)
= \; 2 \,(t^2 \, (u^2 \; + \; v^2)^2)
= \; 2 \, b^2
 

Perimeter where   t = 1 :

a \; + b \; + \; c
= (u^2 + 2 \, u \, v - v^2) \; + \; (u^2 + v^2) \; + \; (u^2 - 2 \, u \, v - v^2)
= 3 \, u^2 \; - \; v^2

OR

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

 
RMS

 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

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

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