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 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 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(x^m − 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" (PDF). Acta Arithmetica LXXXIII (4): 381–389. http://matwbn.icm.edu.pl/ksiazki/aa/aa83/aa8345.pdf. 
• 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.