2 Significant expansion

This is a well-investigated Diophantine equation known as Nagell--Ljunggren equation (they investigated this equation in the 1920s and 1940s, resp). I believe Indeed, it is still open (though conjectured that the information I give three solutions mentioned by the questioner are the only ones; however, it is from a 15 year old papernot even known that the number of solutions is finite, due though there are numerous partial result.

First, so not completely up I restate the question to date)bring the notation in line with some sources I quote.

Early work

What are the solutions $(x,y,n,q)$ of the equation$$\frac{x^n - 1}{x - 1} = y^q$$with integers $x,y>1$, $n>1$, $q \ge 2$ ?

As mentioned, in the question, one finds three 'small' solutions$(3,11,5,2)$, $(7,20,4,2)$, and $(18,7,3,3)$.And, the remaining question isdue to Lunggren :

(1943), who determined all A) Are these three solutions for all the case solutions ?

Or more modestly

(B) Is the sum number of solutions finite?

As said, even (B) is a squareopen; but (A) is conjectured to be true.

By early works of Nagell and Ljunggren it is known that with any of the following conditions there are no other solutions: $q=2$, $n$ a multiple of $3$, $n$ a multiple of $4$, or ($q=3$ and $n$ not $5$ modulo $6$).

There are various further partial numerous additional resultsknown. See, imposing various conditions on $x,y,n$ or $q$ (due to Bennet, Bugeaud, Le, Mignotte, and others) for a survey of the state of the art around a decade ago see, e.g., a 2002 survey (in French) of Bugeaud and Mignotte (which was also the main bases for the above mentioned paperwritten part) available here.There

The early results were obtained via passing to certain rings of algebraic integers; later results often used Baker's method (linear forms in logarithms) and results on Diophantine approximation.Some years ago, the solution of Catalan's conjecture (which is also on a lot somewhat similar equation), by Mihailescu that (as far as I understand, very surprisingly) avoided all these types of recent work tools and used instead (only) results on this type cyclotomic fields/integers, provided a new impetus.

Specifically, it is now known, see Bugeaud and Mihailescu (2007), that

a. for any other solution (so not one of equation three known ones) the smallest prime divisor of $n$ is at least $29$ and $n$ has at most $4$ prime divisors (counted with multiplicity). Moreover, $n$ is prime if $q=3$. And, if $q\mid n$, then $q=n$.

b. to prove that there are no other solutions, it suffices to show that there is no solution with $n\ge 5$ an extra parameter odd prime and other variations); look for Nagell-Ljunggren equation$q$ an odd prime.

In

Moreover, related to the latter assertion Mihailescu recently proved (see here and here) various results in the case there is interest I can somewhat expand/update this answer that $n$ and $q$ are odd primes (saying, in half one of the abstracts that methods used in the cyclotomic approach to FLT are used, so Yemon Choi's intuition was very right).

This answer does certainly not give a daycomplete picture (when in this format, it would be difficult to give one, and no matter the format, it would be impossible for me); I have better access am aware of various omissions I made, and I am afraid there are many of which I am not aware. The references I mentioned should however allow to publications)retrieve more complete information.

[Note: in case the tex is broken, it is not carelessness; at the moment, for technical reasons, I cannot test it myself.]

1

This is a well-investigated Diophantine equation. I believe it is still open (though the information I give is from a 15 year old paper, due to Yu and Le , so not completely up to date).

Early work is due to Lunggren (1943), who determined all solutions for the case the sum is a square. Shorey and Tijdeman proved (1976), among others, that for each r there are only finitely many solutions. And conjectured that there are only finitely many solutions in total.

There are various further partial results known. See, e.g., the above mentioned paper. There is also a lot of recent work on this type of equation (with an extra parameter and other variations); look for Nagell-Ljunggren equation.

In case there is interest I can somewhat expand/update this answer in half a day (when I have better access to publications).