If I remember correctly, it is something like: Prove that the equation x^3 + 2 y

Question

If I remember correctly, it is something like:
Prove that the equation x^3 + 2 y^3 = 7 (u^3 + 2 v^3) has no nontrivial integral solution.
Of course, the trivial solution is when everything is zero.
You need to consider what the possible values are for a perfect cube n^3 (mod 7). Then apply the method of infinite descent. If I remember correctly, it is something like:
Prove that the equation x^3 + 2 y^3 = 7 (u^3 + 2 v^3) has no nontrivial integral solution.
Of course, the trivial solution is when everything is zero.
You need to consider what the possible values are for a perfect cube n^3 (mod 7). Then apply the method of infinite descent. If I remember correctly, it is something like:
Prove that the equation x^3 + 2 y^3 = 7 (u^3 + 2 v^3) has no nontrivial integral solution.
Of course, the trivial solution is when everything is zero.
You need to consider what the possible values are for a perfect cube n^3 (mod 7). Then apply the method of infinite descent.

Explanation / Answer

Claim: x^3 + 2 y^3 = 7 (u^3 + 2 v^3) has no nontrivial integral solution.

Proof: This equation is easily dealth with. No need for any kind of descent.

Read the equation modulo 7. Then

                                                          x^3 + 2 y^3 = 0 (mod 7)

This implies -2 (or 5) is a cube modulo 7.

But the cubes modulo 7 are {k3 mod 7, k=0,1,2,3,4,5,6} = { 0,1,1,-1,1,-1,-1].

So -2 (or 5) is not a cube modulo 7.

conssequantly

                                x^3 + 2 y^3 = 0 (mod 7)

does not have non trivial integer solutions, hence the given equation doesnt have any nontrivial integer solutions.

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