Goormaghtigh conjecture

{{harvtxt|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. {{harvtxt|Nesterenko|Shorey|1998}} showed that, if $m-1=dr$ and $n-1=ds$ with $d\; \backslash ge\; 2$, $r\; \backslash ge\; 1$, and $s\; \backslash ge\; 1$, then $\backslash max(x,y,m,n)$ is bounded by an effectively computable constant depending only on $r$ and $s$. {{harvtxt|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.

{{harvtxt|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, and .

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.

• Feit–Thompson conjecture

• Goormaghtigh, Rene. L’Intermédiaire des Mathématiciens 24 (1917), 88
• {{cite journal | first1=Y. | last1=Bugeaud | first2=T.N. | last2=Shorey | title=On the diophantine equation $\backslash tfrac\{x^m-1\}\{x-1\}=\backslash tfrac\{y^n-1\}\{y-1\}$ | journal=Pacific Journal of Mathematics | volume=207 |issue=1 | year=2002 | pages=61–75 | doi=10.2140/pjm.2002.207.61 | url=http://msp.org/pjm/2002/207-1/pjm-v207-n1-p04-s.pdf | doi-access=free }}
• {{cite journal | first1=R. | last1=Balasubramanian | author1-link=Ramachandran Balasubramanian | first2=T.N. | last2=Shorey | title=On the equation $a(x^\{m\}-1)/(x-1)=b(y^\{n\}-1)/(y-1)$ | journal=Mathematica Scandinavica | volume=46 | year=1980 | pages=177–182 | zbl=0434.10013 | doi=10.7146/math.scand.a-11861|mr=0591599| doi-access=free }}
• {{cite journal | first1=H. | last1=Davenport | first2=D. J. | last2=Lewis | first3=A. | last3=Schinzel | title=Equations of the form $f(x)=g(y)$ | journal=Quad. J. Math. Oxford | volume=2 | year=1961 | pages=304–312 | doi=10.1093/qmath/12.1.304 | mr=0137703 }}
• {{cite book|first=Richard K. |last=Guy|author-link=Richard K. Guy|title=Unsolved Problems in Number Theory | edition=3rd | publisher=Springer-Verlag|date=2004|page=242|isbn=0-387-20860-7 | zbl=1058.11001}}
• {{cite journal | first1=Bo | last1=He | first2=Alan | last2=Togbé | title=On the number of solutions of Goormaghtigh equation for given $x$ and $y$ | journal=Indag. Math. |series=New Series | volume=19 | year=2008 | pages=65–72 | doi=10.1016/S0019-3577(08)80015-8 | mr=2466394 | doi-access=free }}
• {{cite journal | first1=Yu. V. | last1=Nesterenko | author-link=Yuri Valentinovich Nesterenko |first2=T. N. | last2=Shorey | title=On an equation of Goormaghtigh | journal=Acta Arithmetica | volume=LXXXIII | issue=4 | year=1998 | pages=381–389 | url=http://matwbn.icm.edu.pl/ksiazki/aa/aa83/aa8345.pdf | zbl=0896.11010 | doi=10.4064/aa-83-4-381-389 | mr=1610565| doi-access=free }}
• {{cite book | last1=Shorey | first1=T.N. | last2=Tijdeman | first2=R. | author2-link=Robert Tijdeman | title=Exponential Diophantine equations | series=Cambridge Tracts in Mathematics | volume=87 | publisher=Cambridge University Press | year=1986 | isbn=0-521-26826-5 | zbl=0606.10011 | pages=203–204}}
• {{cite journal | first=Pingzhi | last=Yuan | title=On the diophantine equation $\backslash tfrac\{x^3-1\}\{x-1\}=\backslash tfrac\{y^n-1\}\{y-1\}$ | journal=J. Number Theory | volume=112 | year=2005 | pages=20–25 | doi=10.1016/j.jnt.2004.12.002 | mr=2131139 | doi-access=free }}

Category:Diophantine equations

Category:Conjectures

Category:Unsolved problems in number theory