Goormaghtigh conjecture


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 and are
and

Partial results

showed that, for each pair of fixed exponents and, this equation has only finitely many solutions. But this proof depends on Siegel's finiteness theorem, which is ineffective. showed that, if and with,, and, then is bounded by an effectively computable constant depending only on and. showed that for and odd, this equation has no solution other than the two solutions given above.
Balasubramanian and Shorey proved in 1980 that there are only finitely many possible solutions to the equations with prime divisors of and lying in a given finite set and that they may be effectively computed.
showed that, for each fixed and, this equation has at most one solution.

Application to repunits

The Goormaghtigh conjecture may be expressed as saying that 31 and 8191 are the only two numbers that are repunits with at least 3 digits in two different bases.