BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//Sabre//Sabre VObject 4.5.8//EN
CALSCALE:GREGORIAN
BEGIN:VTIMEZONE
TZID:Europe/Zurich
X-LIC-LOCATION:Europe/Zurich
TZURL:http://tzurl.org/zoneinfo/Europe/Zurich
BEGIN:DAYLIGHT
TZOFFSETFROM:+0100
TZOFFSETTO:+0200
TZNAME:CEST
DTSTART:19810329T020000
RRULE:FREQ=YEARLY;BYMONTH=3;BYDAY=-1SU
END:DAYLIGHT
BEGIN:STANDARD
TZOFFSETFROM:+0200
TZOFFSETTO:+0100
TZNAME:CET
DTSTART:19961027T030000
RRULE:FREQ=YEARLY;BYMONTH=10;BYDAY=-1SU
END:STANDARD
END:VTIMEZONE
BEGIN:VEVENT
UID:news317@dmi.unibas.ch
DTSTAMP;TZID=Europe/Zurich:20211104T143247
DTSTART;TZID=Europe/Zurich:20181004T141500
SUMMARY:Number Theory Seminar: Dijana Kreso (TU Graz)
DESCRIPTION:In my talk I will present results that come from a joint work w
 ith M. Bennett and A. Gherga from The University of British Columbia. We s
 tudied Goormaghtigh's equation:\\begin{equation}\\label{eq}\\frac{x^m-1}{x
 -1} = \\frac{y^n-1}{y-1}\, \\\; \\\; y>x>1\, \\\; m > n > 2. \\end{equatio
 n}There are two known solutions $(x\, y\,m\, n)=(2\, 5\, 5\, 3)\, (2\, 90\
 , 13\, 3)$ and it is believed that these are the only solutions. It is not
  known if this equation has finitely or infinitely many solutions\, and no
 t even if that is the case if we fix one of the variables. It is known tha
 t there are finitely many solutions if we fix any two variables. Moreover\
 , there are effective results in all cases\, except when the two fixed var
 iables are the exponents $m$ and $n$. If the fixed $m$ and $n$ additionall
 y satisfy  $\\gcd(m-1\, n-1)>1$\, then there is an effective finiteness r
 esult. My co-authors and me showed that if $n \\geq 3$ is a fixed integer\
 , then there exists an effectively computable constant $c (n)$ such that $
 \\max \\{ x\, y\, m \\} < c (n)$ for all $x\, y$ and $m$ that satisfy Goor
 maghtigh's equation with $\\gcd(m-1\,n-1)>1$.  In case $n \\in \\{ 3\, 4\
 , 5 \\}$\, we solved the equation completely\, subject to this non-coprima
 lity condition.
X-ALT-DESC:In my talk I will present results that come from a joint work wi
 th M. Bennett and A. Gherga from The University of British Columbia. We st
 udied Goormaghtigh's equation:<br />\\begin{equation}\\label{eq}<br />\\fr
 ac{x^m-1}{x-1} = \\frac{y^n-1}{y-1}\, \\\; \\\; y&gt\;x&gt\;1\, \\\; m &gt
 \; n &gt\; 2. <br />\\end{equation}<br />There are two known solutions $(x
 \, y\,m\, n)=(2\, 5\, 5\, 3)\, (2\, 90\, 13\, 3)$ and it is believed that 
 these are the only solutions. It is not known if this equation has finitel
 y or infinitely many solutions\, and not even if that is the case if we fi
 x one of the variables. It is known that there are finitely many solutions
  if we fix any two variables. Moreover\, there are effective results in al
 l cases\, except when the two fixed variables are the exponents $m$ and $n
 $. If the fixed $m$ and $n$ additionally satisfy&nbsp\; $\\gcd(m-1\, n-1)&
 gt\;1$\, then there is an effective finiteness result. My co-authors and m
 e showed that if $n \\geq 3$ is a fixed integer\, then there exists an eff
 ectively computable constant $c (n)$ such that $\\max \\{ x\, y\, m \\} &l
 t\; c (n)$ for all $x\, y$ and $m$ that satisfy Goormaghtigh's equation wi
 th $\\gcd(m-1\,n-1)&gt\;1$.&nbsp\; In case $n \\in \\{ 3\, 4\, 5 \\}$\, we
  solved the equation completely\, subject to this non-coprimality conditio
 n. 
DTEND;TZID=Europe/Zurich:20181004T151500
END:VEVENT
END:VCALENDAR