An old and new approach to Goormaghtigh’s equation

Abstract

We show that if n ≥ 3 n \geq 3 is a fixed integer, then there exists an effectively computable constant c ( n ) c (n) such that if x x , y y , and m m are integers satisfying x m − 1 x − 1 = y n − 1 y − 1 , y > x > 1 , m > n , \begin{equation*} \frac {x^m-1}{x-1} = \frac {y^n-1}{y-1}, \; \; y>x>1, \; m > n, \end{equation*} with gcd ( m − 1 , n − 1 ) > 1 \gcd (m-1,n-1)>1 , then max { x , y , m } > c ( n ) \max \{ x, y, m \} > c (n) . In case n ∈ { 3 , 4 , 5 } n \in \{ 3, 4, 5 \} , we solve the equation completely, subject to this non-coprimality condition. In case n = 5 n=5 , our resulting computations require a variety of innovations for solving Ramanujan-Nagell equations of the shape f ( x ) = y n f(x)=y^n , where f ( x ) f(x) is a given polynomial with integer coefficients (and degree at least two), and y y is a fixed integer.

Goormaghtigh's equation: small parameters

“…As it transpires, this condition allows one to apply a wide variety of effective methods from Diophantine Approximation to the problem, including lower bounds for linear forms in logarithms, Runge's method and the hypergeometric method of Thue and Siegel. Combining results from [4] and [21], we have Theorem 7 (Nesterenko-Shorey, Bennett-Gherga-Kreso). If there is a solution in integers x, y, n and m to equation (1), satisfying (17), then…”

“…On the other hand, since n ≥ 18, we may readily show that 6n 3 − 7n 2 + 2n 24(n − 1) 4 x 0 > 1 6(n − 1) 3 for j ∈ {1, 2}, and that 6n 3 − 7n 2 + 2n 24(n − 1) 4 x 0 < 1 6(n − 1) 3 for j = 3 and n ≥ 43, in each case contradicting (30). Finally, if n = 25 and x 0 = 919, we simply check that equation (1) fails to be satisfied.…”

“…For more recent work, the reader may consult [11], [12], [17], [18], [19] and [30]. The current state of the art for applying techniques from Diophantine Approximation to (1) can be found in [4], and in the papers of Nesterenko and Shorey [21], and of Bugeaud and Shorey [7]. In the last of these, by way of example, one finds the following result.…”

“…is conjectured to have precisely the solutions (x, y, m, n) ∈ { (2,5,5,3), (2,90,13,3)} (2) in integers. Unconditionally, however, the number of such solutions is not known to be finite, even if one fixes one of the variables x, y or n (though recent work of the first author, Gherga and Kreso [4] establishes such a result for a given n under the additional assumption that gcd(m − 1, n − 1) > 1). If one fixes any two of the variables x, y, m or n, however, then the number of solutions to (1) is, in fact, finite.…”

A Note on the Goormaghtigh Equation Concerning Difference Sets

“…The above conjecture is a very difficult problem in Diophantine equations. It was solved for some special cases (see [3, 5, 6, 9, 10, 12, 14–18, 22, 23, 26, 29–37]). But, in general, the problem is far from solved.…”