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Compute the number of ordered triples of integers (x, y, z), 1729 < x, y, z < 1999 which satisfy: x^2 + xy + y^2 = y^3 - x^3 and yz + 1 = y^2 + z. 

 Apr 7, 2020
 #1
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y^3 - x^3 = (y-x)(y^2 + xy + x^2)

 

So

 

(y - x) ( y^2  + xy + x^2)   =  (y^2  + xy + x^2)

 

(y - x)   =  1

 

y = x + 1

 

And

 

yz + 1  = y^2 + z

 

(x + 1)z + 1  = (x+1)^2  + z

 

xz + z + 1  = x^2 + 2x + 1  + z

 

x^2 + 2x - xz  =  0

 

x ( x + 2 - z)  =  0

 

x = 0   or     x + 2   = z

 

So

 

{x , y , z}   =   { x, x + 1, x + 2 }

 

The number of ordered triples  =   {1730, 1731, 1732} .....( 1996, 1997, 1998}  = 1996 - 1730 +  1  =  267

 

 

cool cool cool

 Apr 7, 2020

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