Problem #939: Let $r\geq 2$. An $r$-powerful number $n$ is one such that...

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Progress

The answer to the third question is yes: Nitaj [Ni95] has proved that there are infinitely many triples of coprime $3$-powerful numbers $a,b,c$ such that $a+b=c$, such as\[2^3\cdot 3^5\cdot 73^3+271^3 = 919^3.\]In Nitaj's construction at least two of $a,b,c$ are perfect cubes. Cohn [Co98] constructed infinitely many such triples, none of which are perfect cubes. An alternative construction was given by Walsh [Wa24].

Euler had conjectured that the sum of $k-1$ many $k$th powers is never a $k$th power, but this is false for $k=5$, as Lander and Parkin [LaPa67] found\[27^5+84^5+110^5+133^5=144^5.\]Cambie has found several examples of the sum of $r-2$ coprime $r$-powerful numbers being itself $r$-powerful. For example when $r=5$\[3^761^5=2^83^{10}5^7+2^{12}23^6+11^513^5.\]Cambie has also found solutions when $r=7$ or $r=8$ (the latter even with the sum of $5$ $8$-powerful numbers being $8$-powerful).