Goormaghtigh conjecture


In mathematics, the Goormaghtigh conjecture is a conjecture in number theory named for the Belgian mathematician René Goormaghtigh about the solutions of the exponential Diophantine equation
with distinct integers larger than one and exponents larger than two.
One convention is and in turn.
The conjecture states that the only such solutions are
and

Representation

The fraction of either side of the conjecture exactly represents a finite geometric series. Indeed, and so, for example,.
As such, the exponential Diophantine equation equates two univariate polynomials, with terms and highest order on the left hand side, and on the right.
Alternatively, by cross-multiplication of the fraction's denominators, the equation is equivalently expressed as
or similar forms.
Taking logs,
where the remainder term is a of a ratio of polynomial expressions.
Given, one has with the remainder in the range.

In terms of repunits

Equating two expressions of the form, the Goormaghtigh conjecture may also be expressed as saying that there are only two numbers that are repunits with at least three digits in two different bases. The number 31 may be represented as 111 in base 5 or as 11111 in base 2, while 8191 is 111 in base 90 or 1111111111111 in base 2.

Partial results

The conjecture has been subject to extensive computer supported solution search, especially in small cases or when the fraction is prime. The latter are referred to as "Goormaghtigh primes".
Such search is aided by various necessary congruence relations implied by the equation as well as analytical bounding results, some of which are noted below.
The list also contains known results concerning the finiteness of solution sets under further conditions. Regarding results with asymmetric variable use, again beware that the alternative convention is also used in the literature.
  • For fixed and, loose upper bounds for can be computed from and, as noted, then equals an integer close to. showed that, for each fixed and, the equation has at most one solution. For fixed, equation has at most 15 solutions, and at most two unless is either odd prime power times a power of two, or in a finite set, in which case there are at most three solutions. Furthermore, there is at most one solution if the odd part of is squareful unless has at most two distinct odd prime factors or is in another finite set. If is a power of two, there is at most one solution except for, in which case there are two known solutions. In fact, here and.
  • For prime divisors of and lying in a given finite set, Balasubramanian and Shorey proved in 1980 that there are only finitely many possible solutions and that these can in principle be effectively computed.
  • showed that, for fixed and, the equation has only finitely many solutions. The proof of this, however, depends on Siegel's finiteness theorem, which is ineffective.
  • showed that in the simplifying situation where the exponents are composed from positive integers as in and with, the value is bounded by an effectively computable constant depending only on and.
  • showed that for and odd, the equation has no solution other than the two known ones at all.