What are the known integer solutions to $a^3+b^3=n^5 $? Found:$3^3+6^3=3^5$ to be smallest positive integeral solution.

Question

I came across an interesting and seemingly rare integer identity:

$$3^3 + 6^3 = 3^5.$$

That is:

$$27 + 216 = 243.$$

This satisfies the equation:

$$a^3 + b^3 = n^5.$$

This seems unusual to me since:

The powers on each side are different;

All terms are small positive integers.

I couldn’t find any mention of this identity elsewhere.

My questions:

1. Are there other known positive integer solutions to?

2. Is there a known classification or infinite family of such identities?

3. Could this be the smallest non-trivial solution to this equation?

Any insights, generalizations, or references would be much appreciated. Thanks!

Answer 1

Here is a special case. The solution you have found, and which is also given as an example in the context, concerns the equation $$ a^3+b^3=a^5. $$ This can be rewritten as $b^3=a^3(a^2-1)$. Suppose we find an integer solution $(x,y)$ for the elliptic curve $E:y^2=x^3+1$. Then we let $a=y$, so that $a^2-1=x^3$. Hence our equation is given by $$ b^3=a^3(a^2-1)=y^3x^3 $$ which has a solution $b=yx$. But the integer points of $y^2=x^3+1$ have been studied here:

Find all integer solutions for $x^3+1=y^2$.

Answer 2

Let $(a,b,n)=(px,qx,x)$ and we get $$p^3+q^3=x^2.$$ According to Cohen's book(Number Theory, Volume 2, p.470), one of the parametric solutions is given \begin{align*} p &= s^4-4s^3t-6t^2s^2-4st^3+t^4,\\ q &= 2s^4+4s^3t+4st^3+2t^4,\\ x &= 3(s^2-t^2)(s^4+2s^3t+6t^2s^2+2st^3+t^4).\\ \end{align*}

Taking $t=-1$ to simplify the result and we get

\begin{align*} a &= 3(s^4+4s^3-6s^2+4s+1)(s^2-1)(s^4-2s^3+6s^2-2s+1),\\ b &= 3(2s^4-4s^3-4s+2)(s^2-1)(s^4-2s^3+6s^2-2s+1),\\ n &= 3(s^2-1)(s^4-2s^3+6s^2-2s+1).\\ \end{align*}

This solution gives infinitely many positive solutions.

(a,b,n)
(269952, 80256, 1824)
(4228245, 2363130, 9765)
(37003392, 27195264, 37152)
(221010405, 191489970, 112245)
(1008283392, 978391296, 287424)