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

[math]\displaystyle{ \frac{x^m - 1}{x-1} = \frac{y^n - 1}{y-1} }[/math]

satisfying [math]\displaystyle{ x\gt y\gt 1 }[/math] and [math]\displaystyle{ n,m\gt 2 }[/math] are

[math]\displaystyle{ \frac{5^3-1}{5-1} = \frac{2^5-1}{2-1} = 31 }[/math]

and

[math]\displaystyle{ \frac{90^3-1}{90-1} = \frac{2^{13}-1}{2-1} = 8191. }[/math]

Partial results

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

Balasubramanian and Shorey proved in 1980 that there are only finitely many possible solutions [math]\displaystyle{ (x,y,m,n) }[/math] to the equations with prime divisors of [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] lying in a given finite set and that they may be effectively computed. (He Togbé) showed that, for each fixed [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math], 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 n is squareful unless n has at most two distinct odd prime factors or n 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}.

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 [math]\displaystyle{ \tfrac{x^m-1}{x-1}=\tfrac{y^n-1}{y-1} }[/math]". Pacific Journal of Mathematics 207 (1): 61–75. doi:10.2140/pjm.2002.207.61. http://msp.org/pjm/2002/207-1/pjm-v207-n1-p04-s.pdf. 
• Balasubramanian, R.; Shorey, T.N. (1980). "On the equation [math]\displaystyle{ a(x^{m}-1)/(x-1)=b(y^{n}-1)/(y-1) }[/math]". Mathematica Scandinavica 46: 177–182. doi:10.7146/math.scand.a-11861. 
• Davenport, H.; Lewis, D. J.; Schinzel, A. (1961). "Equations of the form [math]\displaystyle{ f(x)=g(y) }[/math]". Quad. J. Math. Oxford 2: 304–312. doi:10.1093/qmath/12.1.304. 
• Guy, Richard K. (2004). Unsolved Problems in Number Theory (3rd ed.). Springer-Verlag. p. 242. ISBN 0-387-20860-7. 
• He, Bo; Togbé, Alan (2008). "On the number of solutions of Goormaghtigh equation for given [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math]". Indag. Math.. New Series 19: 65–72. doi:10.1016/S0019-3577(08)80015-8. 
• Nesterenko, Yu. V.; Shorey, T. N. (1998). "On an equation of Goormaghtigh". Acta Arithmetica LXXXIII (4): 381–389. doi:10.4064/aa-83-4-381-389. http://matwbn.icm.edu.pl/ksiazki/aa/aa83/aa8345.pdf. 
• 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. 
• Yuan, Pingzhi (2005). "On the diophantine equation [math]\displaystyle{ \tfrac{x^3-1}{x-1}=\tfrac{y^n-1}{y-1} }[/math]". J. Number Theory 112: 20–25. doi:10.1016/j.jnt.2004.12.002.