(x,y,z) positive integers, √x + √y = √z, z ≤ 1000

 
 

x, \; y ,   and   z   are positive integers such that

\sqrt { \, x \, } \; + \; \sqrt { \, y \, } \; = \; \sqrt { \, z \, }
 
 
Find all possible values of   z \; \leq \; 1000

A pattern will emerge.   Find it.

 
 
Solution:

 
x \, y   must be a square.

z \; = \; x \; + \; y \; + \; 2 \, \sqrt { \,x \, y \,}

In general,   x \; = \; k,   y \; = \; k \, n^2,   and   z \; = \; k(n+1)^2

\sqrt { \,k \,} \; + \; \sqrt { \,k \, n^2 \,} \; = \; \sqrt { \,(k \, (n+1)^2 \,}

k \; > \; 0,   n \; \geq \; 0

 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

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

One Response to (x,y,z) positive integers, √x + √y = √z, z ≤ 1000

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