Finding the Number of Ordered Pairs!

Number Theory Level 3

3 y 2 = x 4 + x \Large{3y^2 = x^4 + x}

How many ordered pairs of positive integers ( x , y ) (x,y) satisfy the above equation?


The answer is 0.

Satyajit Mohanty by Satyajit Mohanty , 24, India

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1 solution

Department 8
Aug 5, 2015

We have 3 y 2 = x ( x 3 + 1 ) 3 = x ( x + 1 ) ( x 2 − x + 1 ) y 2 3y^{2} = x(x^{3}+1) \\ 3=\frac{x (x+1)(x^{2}-x+1)}{y^{2}}

We see that at any value of x x the above equation will have non-perfect square and even it has then our answer will not be 3 3 at any y y . Answer : 0 \boxed{0} .

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