Problem 4

Prove that if

n

is a positive integer such that

x^3-3xy^2+y^3=n

has an integer solution

(x,y)

, then it has at least three such solutions. Show that the equation has no integer solutions for

n=2891

Step 2 of 3: Generate a three-cycle of solutions

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In plain words

A finite-order symmetry turns one solution into an orbit of size three.

(x,y)\mapsto(k,-x)\mapsto(-x-k,-k)\mapsto(x,y)

Detailed analysis

The rewritten equation shows $(k,-x)$ is also a solution. Applying the same rule again gives $(-x-k,-k)$ , and a third application returns to $(x,y)$ . If two of these were equal, all coordinates would be zero, forcing $n=0$ , impossible; hence they are distinct for positive $n$ .