Special Sum & Product

Number Theory Level 4

Let x x and y y be integers, satisfying ( x y ) 2 = ( x + y ) 3 − 13 (xy)^2 = (x+y)^3 - 13 .

Compute x + y x+y .


The answer is 17.

Worranat Pakornrat by Worranat Pakornrat , 36, Thailand

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1 solution

Mark Hennings
Jan 14, 2018

According to Bennet & Gahdermarzi , the only integer solutions of the Mordell equation u 2 = v 3 − 13 u^2 = v^3 - 13 are ( u , v ) = ( 70 , 17 ) (u,v) = (70,17) and ( − 70 , 17 ) (-70,17) . Thus the only possible solution to the equation here is { x , y } = 7 , 10 \{x,y\} ={7,10} , and hence x + y = 17 x+y=\boxed{17} .

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