Goormaghtigh conjecture

In mathematics, the Goormaghtigh conjecture is a conjecture in number theory named for the Belgian mathematician René Goormaghtigh. It concerns the exponential Diophantine equation

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

satisfying ${\displaystyle x>y>1}$ and ${\displaystyle m>2}$, and in turn ${\displaystyle n>m>2}$.

The conjecture states that the only non-trivial integer solutions are

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

and

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

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 ${\displaystyle m-1=d\,r}$ and ${\displaystyle n-1=d\,s}$ 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 ${\displaystyle s}$.

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 ${\displaystyle y<2^{2^{y}}}$.

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

By cross-multiplication, the exponential Diophantine equation conjecture may also be expressed as

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

The fraction of either side of the equation exactly represents a geometric series. Indeed, ${\displaystyle \textstyle \sum _{k=0}^{m-1}x^{k}={\frac {x^{m}-1}{x-1}}}$ and, say, ${\displaystyle 1+5+25=31}$. In turn, 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 three digits in two different bases.

• Feit–Thompson conjecture

• Goormaghtigh, Rene. L’Intermédiaire des Mathématiciens 24 (1917), 88
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• 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.