In mathematics, the Goormaghtigh conjecture is a conjecture in number theory named for the Belgian mathematician René Goormaghtigh.
The conjecture is that the only non-trivial integer solutions of the exponential Diophantine equation satisfying
are and Davenport, Lewis & Schinzel (1961) showed that, for each pair of fixed exponents
, this equation has only finitely many solutions.
But this proof depends on Siegel's finiteness theorem, which is ineffective.
Nesterenko & Shorey (1998) showed that, if
is bounded by an effectively computable constant depending only on
Yuan (2005) showed that for
Balasubramanian and Shorey proved in 1980 that there are only finitely many possible solutions
to the equations with prime divisors of
lying in a given finite set and that they may be effectively computed.
He & Togbé (2008) showed that, for each fixed
For fixed x (or y), equation has at most 15 solutions, and at most two unless x is either odd prime power times a power of two, or in the finite set {15, 21, 30, 33, 35, 39, 45, 51, 65, 85, 143, 154, 713}, in which case there are at most three solutions.
Furthermore, there is at most one solution if the odd part of x is squareful unless x has at most two distinct odd prime factors or x is in a finite set {315, 495, 525, 585, 630, 693, 735, 765, 855, 945, 1035, 1050, 1170, 1260, 1386, 1530, 1890, 1925, 1950, 1953, 2115, 2175, 2223, 2325, 2535, 2565, 2898, 2907, 3105, 3150, 3325, 3465, 3663, 3675, 4235, 5525, 5661, 6273, 8109, 17575, 39151}.