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

(x ^{m - 1})/(x-1)= (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 with at least 3 digits 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) 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

References

* R. Balasubramanian; T.N. Shorey (1980). "On the equation a(xm-1)/(x-1) = b(yn-1)/(y-1)". Math. Scand. 46: 177–182.
* Richard K. Guy (2004). Unsolved Problems in Number Theory. Springer-Verlag. p. 102. ISBN 0-387-20860-7.
* Yu. V. Nesterenko; T. N. Shorey (1998). "On an equation of Goormaghtigh". Acta Mathematica LXXXIII (4): 381–389.
* T.N. Shorey; R. Tijdeman (1986). Exponential Diophantine equations. Cambridge Tracts in Mathematics. 87. Cambridge University Press. pp. 203–204. ISBN 0-521-26826-5.

Number Theory