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

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

and

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

Partial results

Davenport, Lewis & Schinzel (1961) showed that, for each pair of fixed exponents $m$ and $n$, 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 $m-1=dr$ and $n-1=ds$ with $d\geq 2$, $r\geq 1$, and $s\geq 1$, then $\max(x,y,m,n)$ is bounded by an effectively computable constant depending only on $r$ and $s$. Yuan (2005) showed that for $m=3$ and odd $n$, this equation has no solution $(x,y,n)$ other than the two solutions given above.

Balasubramanian and Shorey proved in 1980 that there are only finitely many possible solutions $(x,y,m,n)$ to the equations with prime divisors of $x$ and $y$ lying in a given finite set and that they may be effectively computed. He & Togbé (2008) showed that, for each fixed $x$ and $y$, this equation has at most one solution. For fixed x (or y), equation has at most 15 solutions, and at most two unless x is either odd prime power times a power of two, or in the finite set {15, 21, 30, 33, 35, 39, 45, 51, 65, 85, 143, 154, 713}, in which case there are at most three solutions. Furthermore, there is at most one solution if the odd part of x is squareful unless x has at most two distinct odd prime factors or x is in a finite set {315, 495, 525, 585, 630, 693, 735, 765, 855, 945, 1035, 1050, 1170, 1260, 1386, 1530, 1890, 1925, 1950, 1953, 2115, 2175, 2223, 2325, 2535, 2565, 2898, 2907, 3105, 3150, 3325, 3465, 3663, 3675, 4235, 5525, 5661, 6273, 8109, 17575, 39151}. If x is a power of two, there is at most one solution except for x=2, in which case there are two known solutions. In fact, max(m,n)<4^x and y<2^(2^x).

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 ${\tfrac {x^{m}-1}{x-1}}={\tfrac {y^{n}-1}{y-1}}$" (PDF). Pacific Journal of Mathematics. 207 (1): 61–75. doi:10.2140/pjm.2002.207.61.
• Balasubramanian, R.; Shorey, T.N. (1980). "On the equation $a(x^{m}-1)/(x-1)=b(y^{n}-1)/(y-1)$". Mathematica Scandinavica. 46: 177–182. doi:10.7146/math.scand.a-11861. MR 0591599. Zbl 0434.10013.
• Davenport, H.; Lewis, D. J.; Schinzel, A. (1961). "Equations of the form $f(x)=g(y)$". Quad. J. Math. Oxford. 2: 304–312. doi:10.1093/qmath/12.1.304. MR 0137703.
• Guy, Richard K. (2004). Unsolved Problems in Number Theory (3rd ed.). Springer-Verlag. p. 242. ISBN 0-387-20860-7. Zbl 1058.11001.
• He, Bo; Togbé, Alan (2008). "On the number of solutions of Goormaghtigh equation for given $x$ and $y$". Indag. Math. New Series. 19: 65–72. doi:10.1016/S0019-3577(08)80015-8. MR 2466394.
• 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.
• Shorey, T.N.; Tijdeman, R. (1986). Exponential Diophantine equations. Cambridge Tracts in Mathematics. Vol. 87. Cambridge University Press. pp. 203–204. ISBN 0-521-26826-5. Zbl 0606.10011.
• Yuan, Pingzhi (2005). "On the diophantine equation ${\tfrac {x^{3}-1}{x-1}}={\tfrac {y^{n}-1}{y-1}}$". J. Number Theory. 112: 20–25. doi:10.1016/j.jnt.2004.12.002. MR 2131139.