A number theory problem by Aly Ahmed

Number Theory Level 3

Find the number of pairs of integers ( x , y ) (x,y) satisfying x 2 + 2 x = y 3 x^2 + 2x = y^3 .


The answer is 5.

Aly Ahmed by Aly Ahmed , 34, Egypt

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2 solutions

Marvin Kalngan
Jun 18, 2020

I did it by trial and error.

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Pavan Kartik
Jun 17, 2020

If we complete the square on the lhs we get (x+1)^2 -1 = y^3 which means that there are 2 perfect powers with a difference of 1. Since 3 and 2 are the only possible values, we can have x=-4 y=2, x=2, y=2 and if we make x=-1, we get y=-1, and for y=0 we can make x= -2 or x=0. In total we have 5 possible solutions

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