MathLabs

Problem 4

Prove that if nn is a positive integer such that x3−3xy2+y3=nx^3-3xy^2+y^3=n has an integer solution (x,y)(x,y), then it has at least three such solutions. Show that the equation has no integer solutions for n=2891n=2891.
Step 1 of 3: Rewrite the form using k=y−xk=y-x
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In plain words

The cubic form hides a three-cycle; the difference of the coordinates reveals it.

k=y−x⟹x3−3xy2+y3=k3−3kx2−x3k=y-x\quad\Longrightarrow\quad x^3-3xy^2+y^3=k^3-3kx^2-x^3
Detailed analysis

Substitute y=x+ky=x+k and expand; the equation becomes k3−3kx2−x3=nk^3-3kx^2-x^3=n.