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

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

satisfying ${\displaystyle x>y>1}$ and ${\displaystyle n,m>2}$ are

${\displaystyle {\frac {5^{3}-1}{5-1}}={\frac {2^{5}-1}{2-1}}=31}$

and

${\displaystyle {\frac {90^{3}-1}{90-1}}={\frac {2^{13}-1}{2-1}}=8191.}$

Partial results

Davenport, Lewis & Schinzel (1961) showed that, for each pair of fixed exponents ${\displaystyle m}$ and ${\displaystyle n}$, this equation has only finitely many solutions. But this proof depends on Siegel's finiteness theorem, which is ineffective. Nesterenko & Shorey (1998) showed that, if ${\displaystyle m-1=dr}$ and ${\displaystyle n-1=ds}$ with ${\displaystyle d\geq 2}$, ${\displaystyle r\geq 1}$, and ${\displaystyle s\geq 1}$, then ${\displaystyle \max(x,y,m,n)}$ is bounded by an effectively computable constant depending only on ${\displaystyle r}$ and ${\displaystyle s}$. Yuan (2005) showed that for ${\displaystyle m=3}$ and odd ${\displaystyle n}$, this equation has no solution ${\displaystyle (x,y,n)}$ other than the two solutions given above.

Balasubramanian and Shorey proved in 1980 that there are only finitely many possible solutions ${\displaystyle (x,y,m,n)}$ to the equations with prime divisors of ${\displaystyle x}$ and ${\displaystyle y}$ lying in a given finite set and that they may be effectively computed. He & Togbé (2008) showed that, for each fixed ${\displaystyle x}$ and ${\displaystyle y}$, this equation has at most one solution.

Application to repunits

The Goormaghtigh conjecture may be expressed as saying that 31 (111 in base 5, 11111 in base 2) and 8191 (111 in base 90, 1111111111111 in base 2) are the only two numbers that are repunits with at least 3 digits in two different bases.

See also

• Feit–Thompson conjecture

References

• Goormaghtigh, Rene. L’Intermédiaire des Mathématiciens 24 (1917), 88
• Bugeaud, Y.; Shorey, T.N. (2002). "On the diophantine equation ${\displaystyle {\tfrac {x^{m}-1}{x-1}}={\tfrac {y^{n}-1}{y-1}}}$" (PDF). Pacific Journal of Mathematics. 207 (1): 61–75.CS1 maint: ref=harv (link)
• Balasubramanian, R.; Shorey, T.N. (1980). "On the equation ${\displaystyle a(x^{m}-1)/(x-1)=b(y^{n}-1)/(y-1)}$". Mathematica Scandinavica. 46: 177–182. doi:10.7146/math.scand.a-11861. MR 0591599. Zbl 0434.10013.CS1 maint: ref=harv (link)
• Davenport, H.; Lewis, D. J.; Schinzel, A. (1961). "Equations of the form ${\displaystyle f(x)=g(y)}$". Quad. J. Math. Oxford. 2: 304–312. doi:10.1093/qmath/12.1.304. MR 0137703.CS1 maint: ref=harv (link)
• Guy, Richard K. (2004). Unsolved Problems in Number Theory (3rd ed.). Springer-Verlag. p. 242. ISBN 0-387-20860-7. Zbl 1058.11001.CS1 maint: ref=harv (link)
• He, Bo; Togbé, Alan (2008). "On the number of solutions of Goormaghtigh equation for given ${\displaystyle x}$ and ${\displaystyle y}$". Indag. Math. N. S. 19: 65–72. doi:10.1016/S0019-3577(08)80015-8. MR 2466394.CS1 maint: ref=harv (link)
• Nesterenko, Yu. V.; Shorey, T. N. (1998). "On an equation of Goormaghtigh" (PDF). Acta Arithmetica. LXXXIII (4): 381–389. doi:10.4064/aa-83-4-381-389. MR 1610565. Zbl 0896.11010.CS1 maint: ref=harv (link)
• Shorey, T.N.; Tijdeman, R. (1986). Exponential Diophantine equations. Cambridge Tracts in Mathematics. 87. Cambridge University Press. pp. 203–204. ISBN 0-521-26826-5. Zbl 0606.10011.CS1 maint: ref=harv (link)
• Yuan, Pingzhi (2005). "On the diophantine equation ${\displaystyle {\tfrac {x^{3}-1}{x-1}}={\tfrac {y^{n}-1}{y-1}}}$". J. Number Theory. 112: 20–25. doi:10.1016/j.jnt.2004.12.002. MR 2131139.CS1 maint: ref=harv (link)