positive integer solutions to $x^3+y^3=3^z$

Question

I am seeking all positive integer solutions to the equation $x^3+y^3=3^z$.

After doing number crunching, I think there are no solutions. But I am unable to prove it.

Attempt

If $x$ and $y$ have common divisor $d$, we have $d^3(m^3+n^3)=3^z$. So $d$ must be a power of $3$, and we are back to where we started. So we assume $x$ and $y$ are coprime.

Testing the parity, we have sum of 2 cubes to be odd. WLOG, we can assume $x$ is even and $y$ is odd.

Trying mod $3$, we have $x+y=0 \pmod 3$. Since $x$ and $y$ are coprime, $x$ and $y$ must be congruent to $1$ and $-1$ or vice-versa.

If I assume $x=3m+1$ and $y=3n-1$, expand out and simplify, I get $27(m^3+n^3)+27(m^2-n^2)+9(m+n)=3^z$. If I assume $z \geq 3$, this gives $(m^3+n^3)+(m^2-n^2)+\frac{m+n}{3}=3^{z-3}$. But I don't see how to proceed.

I also tried mod $9$ but didn't get anywhere, it didn't cut down the possibilities by much.

I also tried letting $y=x+r$. Then \begin{align*} x^3+y^3 &= x^3+(x+r)^3 \\ &= x^3 + (x^3+3x^2r+3xr^2+r^3) \\ &= 2x^3+3x^2r+3xr^2+r^3 \\ &= 3^z \end{align*}

Then $3\mid 2x^3+3x^2r+3xr^2+r^3$, and $3\mid 3x^2r+3xr^2$, so this implies $3 \mid 2x^3+r^3$. But this doesn't yield any contradiction.

Can anyone supply a proof? Or if my hypothesis is wrong, how to derive all the integer solutions?

Thank you.

Answer 1

$27(m^3+n^3) +27(m^2-n^2) + 9(m+n) = 3^z$

$(m+n)(3m^2 +3n^2-3mn+3m-3n+1) = 3^{z-2}$

since $z \ne 2$, we have $3(m^2 +n^2-mn+m-n)+1$ divides a power of $3$ or is equal to $1$, both of which is not possible.

Answer 2

You already proved that we can assume that $x,y$ are relatively prime.

We can find easly solution for $n\leq 2$. Let $n\geq 3$.

So $x+y = 3^a$ and $x^2-xy+y^2=3^b$ for some nonegative integers $a+b=n$.

If $b\geq 2$ then $9\mid x^2-xy+y^2$ and $3\mid x+y$, so $$9\mid (x+y)^2-(x^2-xy+y^2)=3xy\implies 3\mid xy$$ A contradiction. So $b\leq 2$ and this should be easly done.

Answer 3

Solution by LTE, let $d=(x,y)>1$, $x=dx_1$, $y=dy_1$, $(x_1,y_q)=1$ then $$d^3\mid x^3+y^3=3^z$$ then $d\mid 3^a$ for a positive integer $a<z/3$. Dividing by $d^3$ we obtain $x^3_1+y^3_1=3^b$, for a positive integer $b=z-a$. This is a equation similar to the original, then we can assume $(x,y)=1$. If $3\nmid x+y$ then $x+y=1$ a contradiction. Then we have all the conditions to use LTE: $$v_3(x^3+y^3)=v_3(x+y)+v_3(3)=v_3(x+y) +1=v_3(3^z)=z$$ From here we conclude that $x+y=3^{z-1}$ this implies that $x^2-xy+y^2=3$, $(x-y)^2+xy=3$, if $|x-y|\ge 2$ we have a contradiction, then there are two cases: Case 1. $|x-y|=0$ this is not possible due $x$ and $y$ are coprimes and $x+y>2$. Case 2. $|x-y|=1$ implying $xy=2$, with solutions $(x,y)=(2,1)$ or $(1,2)$, implying $3^{z-1}=2+1\Rightarrow z=2$.

Then the solutions to the original equation are: $(x,y,z)=(3^k,2\cdot 3^k,3k+2)$ or $(2\cdot 3^k, 3^k,3k+2)\forall k\in \mathbb{N}$.