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

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

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 that are repunits with at least 3 digits in two different bases.

Balasubramanian and Shorey proved in 1980 that there are only finitely many possible solutions to the equations in (x,y,m,n) with prime divisors of x and y lying in a given finite set and that they may be effectively computed.

See also

• Feit–Thompson conjecture
• Nagell–Ljunggren equation

References

• Goormaghtigh, Rene. L’Intermédiaire des Mathématiciens 24 (1917), 88