Goormaghtigh conjecture

In mathematics, the Goormaghtigh conjecture is a conjecture in number theory named for the mathematician René Goormaghtigh. The conjecture is that the only non-trivial integer solutions of the exponential Diophantine equation

:$frac\{x^m\; -\; 1\}\{x-1\}=frac\{y^n\; -\; 1\}\{y\; -\; 1\}$

satisfying "x", "y" > 1 and "n", "m" > 2 are
* ("x", "y", "m", "n") = (5, 2, 3, 5); and
* ("x", "y", "m", "n") = (90, 2, 3, 13).

This may be expressed as saying that 31 and 8191 are the only two numbers which are repunits in two different bases.

Balasubramanian and Shorey have proved that there are only finitely many possible solutions to the equations in ("x","y","m","n") and that they may be effectively computed.

ee also

*Feit-Thompson conjecture

References

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