Which numbers are both a power of 3 and a cube?

Which numbers are both a power of 1/3 and a cube root?

@christianp The numbers of the form $3^{3j}$ for $j\in \mathbb N$ and the the numbers of the form $1/3^j$ for $j\in\mathbb N$ I guess?

@christianp I only consider positive power here...

@christianp Let $0 < k, p$:
$pᵏ = kᵖ$$k \log p = p \log k$$\frac {\log p} p = \frac {\log k} k$
A trivial solution is when $k = p$, e.g. when $p := 3$ and $p := 1/3$. What about the other?

Let $f : R₊ → R₊$, $f(k) := \frac {\log k} k$. It is smooth but not monotonic.
$f'(k) = \frac 1 {k²} (1 – \log k) = 0$
Here $k = \exp 1$ is the maximum, and $f(x) = C$ doesn't have a solution for $\frac 1 {\exp 1} < C$, has just one for $C ≤ 0$, and has two for the rest…

@christianp For $\frac 1 {\exp 1} = C$, there is also just one solution.

When $p := 1/3$, $f(p)$ is negative, so the trivial solution is the only.
When $p := 3$, $f(p)$ is in the region where there is also the second solution.
Generally, for large $p > 1$, there must be one large and one small $k$. Not sure how to express the latter precisely.

Oh that is fun... You lead me to the Lambert's function... and an approximate solution ... via wolfram...

@kat @christianp Thanks for the reference!
The W function is the proper name of what I was looking for:

en.wikipedia.org/wiki/Lambert_

@amiloradovsky why did you take $k$ to be the same in both cases?
$3^6 = 9^3$ is both a power of 3 and a cube.

@christianp
I misunderstood the question.
The generalized problem would then be to find all $k$ and $l$ for a fixed $p$, such that $0 < k, l, p$ and $pˡ = kᵖ$. Then
$\frac {\log p} p = \frac {\log k} l$$l = \frac {\log k} P$$k = \exp (P l)$
where $P := \frac {\log p} p$ and all the snolutions may be obtained by varying either $k$ or $l$.

This does not fully describe the problem as stated. For the first part you have here described "a number that is both a pth power of 3 and the cube of p"; to account for all numbers which are both powers of 3 and cubes required introducing two variables, as $r = 3ˢ = t³$.

@kimreece
Yes, @christianp already pointed it out (see above). And, with the two variables, the problem is much simpler, and less interesting.
WRT the notation, I actually used $p$ to generalize $3$, the given data, to indicate that it's positive. While for the coefficients to be found — $k$ and $l$, to indicate that these don't have to be integers.
However, if I'm not mistaken, e.g. assumes all the letters from 'i' to 'n' to be integers…

@christianp In which domain are these numbers? If all of R is allowed then it is underdetermined and you get an infinite number of solutions. Do you prefer restriction to Q, or are number fields suitable?

@kimreece @christianp
Number field is the terminology to distinguish the use of the word in algebra and in e.g. agriculture/civil engineering? :)
…or section of vector bundle…

One interesting generalization is: given a type $A$, say, a finite set, or an algebraic structure, find such types $X, Y$ that
$X→A ≃ A→Y$
That is the spaces of functions / maps / homomorphisms are isomorphic.

@kimreece up to you. That's part of what I'm getting at. I don't have a particular answer in mind

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