{x+y, x^2+y^2, x^3+y^3} are squares, and {x^4+y^4} cubes

 
 
Find two positive integers such that

x \; + \; y \; = \; A^2
x^2 \; + \; y^2 \; = \; B^2
x^3 \; + \; y^3 \; = \; C^2
x^4 \; + \; y^4 \; = \; D^3

 

Here’s one solution:

 
(x, \; y) \; = \; (546490752921705, \; 291461734891576)
 

546490752921705 \; + \; 291461734891576 \; = \; 28947409^2

546490752921705^2 \; + \; 291461734891576^2 \; = \; 619356186644599^2

546490752921705^3 \; + \; 291461734891576^3 \; = \; 13710225827603534330699^2

546490752921705^4 \; + \; 291461734891576^4 \; = \; 45853598085630549601^3

 
 
Find other solutions.

 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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