How many solutions?

Number Theory Level 3

Find the number of integer solutions $(x,y)$ satisfying $(x+2)^4 - x^4 = y^3$ .

Five solutions No solution Two solutions One solution

1 solution

Bloons Qoth
Jun 19, 2016

${(x+2)^4-x^4=y^3}$
${8x^3+24x^2+32x+16=y^3}$
${8(x^3+3x^2+4x+2)=y^3}$
If $y=2n$ .........................................................................substitude
${x^3+3x^2+4x+2=n^3}$
If $x \geq0$
${\sqrt[3]{ (x+1)^3<(x^3+3x^2+4x+2=n^3)<(x+2)^3} }$
$x+1<n<x+2$
By contradiction, no $\mathbb{Z}$ solutions exists for $n$
Therefore, $x \ngeq 0$
.
If $x \leq -1$
We assume $(-1,0)$ is the solution
.
If $x \leq -2$
$k, t \in \mathbb{Z}$
${(k+4)^4-k^4=t^3}$
${k_2=-k-2}, {t_2=-t}$ ...............................................substitude
${k_2^4-(k_2+2)^4=t_2^3}$
Therefore, $x\nleq -2$
.
So $(-1,0)$ is the only solution $\blacksquare$