Goormaghtigh conjecture

In mathematics, the Goormaghtigh conjecture is a conjecture in number theory named for the Belgian mathematician René Goormaghtigh about the solutions of the exponential Diophantine equation

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

with distinct integers ${\displaystyle x,y}$ larger than one and exponents larger than two. One convention is ${\displaystyle x>y>1}$ and in turn ${\displaystyle n>m>2}$.

The conjecture states that the only such solutions are

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

and

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

Representation

The fraction of either side of the conjecture exactly represents a geometric series. Indeed, ${\displaystyle \textstyle {\frac {x^{m}-1}{x-1}}=\sum _{k=1}^{m}x^{k-1}}$ and so, for example, ${\displaystyle 31=1+5+25=5^{0}+5^{1}+5^{2}}$. As such, the exponential Diophantine equation equates two univariate polynomials, with ${\displaystyle m}$ terms and highest order ${\displaystyle x^{m-1}}$ on the left hand side, and ${\displaystyle n>m}$ on the right.

Alternatively, by cross-multiplication of the fraction's denominators, the equation is equivalently expressed as

${\displaystyle x^{m}+x\cdot y^{n}+y=y^{n}+y\cdot x^{m}+x,}$

or similar forms.

In terms of repunits

Equating two expressions of the form ${\displaystyle \textstyle \sum _{k=1}^{m}{\textit {1}}\cdot x^{k-1}}$, the Goormaghtigh conjecture may also be expressed as saying that there are only two numbers that are repunits with at least three digits in two different bases. The number 31 may be represented as 111 in base 5 or as 11111 in base 2, while 8191 is 111 in base 90 or 1111111111111 in base 2.

Partial results

Yuan (2005) showed that for ${\displaystyle m=3}$ and odd ${\displaystyle n}$, the equation has no solution ${\displaystyle (x,y,n)}$ other than the two solutions given above.

The conjecture has been subject to extensive computer supported solution search, especially in small cases (restricting to ${\displaystyle x}$ in the thousands, or alternatively restricting to ${\displaystyle x^{m}}$ with around a dozen digits) or when the fraction is prime (hundreds of digits). This is aided by various necessary congruence relations. For fixed ${\displaystyle x}$ and ${\displaystyle y}$, loose upper bounds for ${\displaystyle n}$ can be computed from ${\displaystyle x}$. Taking logs relates the exponents as ${\displaystyle {\tfrac {m}{n}}={\tfrac {\ln y}{\ln x}}+O\left({\tfrac {1}{n}}\right)}$.

Nesterenko & Shorey (1998) showed that if the exponents are composed from positive integers as in ${\displaystyle m-1=d\,r}$ and ${\displaystyle n-1=d\,s}$ with ${\displaystyle d>1}$, then ${\displaystyle \max(x,y,m,n)}$ is bounded by an effectively computable constant depending only on ${\displaystyle r}$ and ${\displaystyle s}$.

Davenport, Lewis & Schinzel (1961) showed that, for each pair of fixed exponents ${\displaystyle m}$ and ${\displaystyle n}$, the equation has only finitely many solutions. The proof of this, however, depends on Siegel's finiteness theorem, which is ineffective.

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. For fixed x (or y), equation has at most 15 solutions, and at most two unless y 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 y is squareful unless y has at most two distinct odd prime factors or y 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 y is a power of two, there is at most one solution except for y = 2, in which case there are two known solutions. In fact, ${\displaystyle \max(m,n)<4^{y}}$ and ${\displaystyle y<2^{2^{y}}}$.

Again, beware that the alternative convention, ${\displaystyle y>x}$, is also used in the literature.

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. doi:10.2140/pjm.2002.207.61.
• 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.
• 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.
• 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 ${\displaystyle x}$ and ${\displaystyle y}$". 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 ${\displaystyle {\tfrac {x^{3}-1}{x-1}}={\tfrac {y^{n}-1}{y-1}}}$". J. Number Theory. 112: 20–25. doi:10.1016/j.jnt.2004.12.002. MR 2131139.