<s>[INST] Number theory: Prime powers and cubes Determine all triples $(p,a,b)$ of positive integers, where $p$ is prime and $a \leq b$  such that $$p^a+p^b$$ is a perfect cube.
I came across this question while looking at past maths Olympiad papers in my country. It has been frustrating me for a while. The only progress I've made is the following - 
Let $p^a+p^b=k^3$.
Then $p^a(1+p^{b-a})=k^3$, but unless $p=2,a=b$, $\gcd(p^a,p^{b-a}+1)=1$.
Case 1: $b=a$, $p=2$.
Then $k^3=2^a+2^a=2^{a+1}$, so $a \equiv 2 \pmod3$. Hence $(2,a,a)$ is a solution $\forall \; \; a \equiv 2 \pmod3$.
Case 2: Case 1 is not true.
Then as $\gcd(p^a,p^{b-a}+1)=1$, $p^a$ and $1+p^{b-a}$ must both be cube and hence $3|a$.
Let $a=3m$,
I can't seem to get beyond this point although I did try ignoring the $p^a$ and focusing on making the other term a cube. I did some factorizing but it didn't help (as far as I could see). 
Thanks in advance for any help.
 [/INST] You took care of the case $a=b$, and obtained  the infinite family of solutions
$2^{3k+2}+2{3k+2}$. 
So assume that $a\lt b$. You showed that for $p^a(1+p^{b-a})$ a cube, $a$ should be divisible by $3$, and $1+p^c$ should be a cube, where $c=b-a$. We com[lete things from there.
If $b-a=1$, we want $1+p$ to be a cube, say $x^3$. The factorization $x^3-1=(x-1)(x^2+x+1)$ tells us that the only possibility is $x=2$, giving $p=7$.
This generates an infinite number of variants, namely 
$$7^{3k}+7^{3k+1}.$$ 
So now we look at the case $1+p^c$ a cube, where $c\gt 1$. As Ross Millikan points out, there is no solution, by Mihailescu's Theorem.   But let's see whether we can prove it without heavyweight machinery.
Suppose $p^c=x^3-1=(x-1)(x^2+x+1)$. Thus each of $x-1$ and $x^2+x+1$ is a power of $p$. Except in the trivial case $x=2$, $p$ must divide each of $x-1$ and $x^2+x+1$. Any common divisor of these must divide $(x^2+x+1)-(x-1)^2$, so it divides $3x$, and therefore $3$.
So we must have $p=3$. Moreover, since each of $x-1$ and $x^2+x+1$ is a proper power of $3$, the only remaining possibility is $x-1=3$. This doesn't work.  
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