The following equation in integers (Goormaghtigh’s equation) is considered: \[ \frac{x^m-1}{x-1}=\frac{y^n-1}{y-1},\; y>x>1,\; m>n>2, \quad \gcd(m-1,n-1)=d>1.\tag{1} \] The first main result of the paper is the following:
Theorem 1. If there is a solution in integers \(x, y, n, m\), to equation (1), then \(x<(3d)^{4n/d}\leq 36^n\). In particular, if \(n\) is fixed, there is an effectively computable constant \(c=c(n)\) such that \(\max\{x,y,m\}<c\).
The authors refine their approach and combine with some new results from computational Diophantine approximation, in order to achieve the complete solution of equation (1) for small fixed values of \(n\).
Theorem 2. The only solutions to (1) with \(n\in\{3,4,5\}\) are \((x,y,m,n)=(2,5,5,3)\) and \((2,90,13,3)\).
From the paper: “Essentially half of the current paper is concerned with developing Diophantine approximation machinery for the case \(n=5\) in Theorem 2. Here, ‘off-the-shelf’ techniques for finding integral points on models of elliptic curves or for solving Ramanujan-Nagell equations of the shape \(F(x)= z^n\) (where \(F\) is a polynomial and \(z\) a fixed integer) do not apparently permit the full resolution of this problem in a reasonable amount of time. The new ideas introduced here are explored more fully in the general setting of Thue-Mahler equations in the forthcoming paper [not yet available][…]
In Section 2, we derive ‘good’ rational approximations to certain algebraic numbers associated to solutions of (1).
Section 3 contains relevant details about Padé approximation to the binomial function. In Sections 4 and 5, we find the proofs of Theorems 1 and 2, respectively. In the latter case, to treat small fixed values of \(n\) and \(x\) in equation (1), we appeal to a variety of techniques from computational Diophantine approximation.
Most interestingly, in case \(n = 5\), we sharpen existing techniques for solving Thue-Mahler equations, and specialize them to our problem.
We note that this section may essentially be read independently of the rest of the paper. For each \(x\), we restrict the problem to that of solving a number of related \(S\)-unit equations, where \(S\) is the set of primes dividing \(x\).
We then generate a large upper bound on the exponents of these equations using bounds for linear forms in logarithms, both Archimedean and non-Archimedean. Finally, unlike traditional examples of Thue-Mahler equations, where extensive use of geometric and \(p\)-adic reduction techniques are typically required, using only a few iterations of the \(LLL\) algorithm, we reduce this bound significantly, after which we apply a naive search to complete our computation. We will, in fact, employ two quite different algorithms for solving Thue-Mahler equations, one for which we must compute the class group of a number field and one which avoids this computation altogether. For a given value of \(x\), one of these versions may be significantly faster than the other; we list some timings for examples to illustrate this difference.”
Section 6 which is technical and long (almost 18 pages), treats the case \(n=5\) when \(x\) is “small”, by reducing the resolution of equation (1) to that of a large number of Thue-Mahler equations. The authors have many computational challenges to overcome not by mere “brute force” of course. For the resolution of the resulting Thue-Mahler equations they need to improve on earlier techniques (for example those due to Tzanakis and de Weger) by using ideas of a forthcoming paper by Gherga, von Känel, Matschke and Siksek.
Section 7 is devoted to the following auxiliary proposition:
If \(\displaystyle{ C(k,d):=d^k\prod_{p|d}p^{\mathrm{ord}_p(k!)}}\), then \(2^k\leq C(k,2)<4^k\) and \(d^k\leq C(k,d)<(2d\log d)^k\) for \(d>2\).
Finally, Section 8 contains a few concluding remarks.