A geometric series equalling a power of an integer

2011-03-17T00:35:03Z

The following problem cropped up whilst considering generalised quadrangles with a product structure, and it boils down to a simple number theoretic problem. Let $s$ be an integer greater than 2 and suppose the geometric series $(s^r-1)/(s-1)$ is a nontrivial power of a positive integer. It seems the following is true:

If $r=3$, then $s= 18$.
If $r=4$, then $s = 7$.
If $r=5$, then $s = 3$.
If $r>5$, there are no solutions.

Does anyone know a proof of this curious property?

Answer by quid for A geometric series equalling a power of an integer

2011-03-17T03:02:37Z

This is a well-investigated Diophantine equation known as Nagell--Ljunggren equation (they investigated this equation in the 1920s and 1940s, resp). Indeed, it is conjectured that the three solutions mentioned by the questioner are the only ones; however, it is not even known that the number of solutions is finite, though there are numerous partial result.

Below, I try to give some rough overview of some results that are known, and some references to (recent) articles.

First, I restate the question to bring the notation in line with some sources I quote.

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 is:

(A) Are these three solutions all the solutions ?

Or more modestly

(B) Is the number of solutions finite?

As said, even (B) is open; 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 $6$).

Shorey and Tijdeman proved (1976) that the number of solutions is finite with any of the following conditions: $x$ is fixed, $n$ has a fixed prime divisor, $y$ has a fixed prime divisor. Also, Shorey proved that the ABC-conjecture implies that the number of solutions is finite.

There are numerous additional results, 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 written part) available here.

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 on a somewhat similar equation), by Mihailescu that (as far as I understand, very surprisingly) avoided all these types of tools and used instead (only) results on 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 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 odd prime and $q$ an odd prime.

Moreover, related to the latter assertion Mihailescu recently proved (see here and here) various results in the case that $n$ and $q$ are odd primes (saying, in 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 complete picture (in this format, it would be difficult to give one, and no matter the format, it would be impossible for me); I 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 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.]

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A geometric series equalling a power of an integer

Answer by quid for A geometric series equalling a power of an integer