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BEGIN:VEVENT
DTSTAMP:20230123T104351Z
DTSTART:20230221T150000Z
DTEND:20230221T160000Z
SUMMARY:Manchester Number Theory Seminar - Paul Voutier
UID:{http://www.columbasystems.com/customers/uom/gpp/eventid/}ou9-ld8nxad
0-sz019m
DESCRIPTION:Speaker: Paul Voutier\n\nTitle: A kit for linear forms in thr
ee logarithms\n(joint work with Maurice Mignotte)\n\nAbstract: Lower bou
nds for linear forms in logarithms are a powerful tool that have found a
pplication to many number theory problems. In fact\, many problems can b
e reduced to linear forms in two or three logarithms. As a result\, good
lower bounds for such linear forms have resulted in the complete soluti
on of several outstanding problems [\;BHV]\;\, [\;BMS]\;.\n\
nIn the early 2000s\, Mignotte produced his "A kit on linear forms in th
ree logarithms" manuscript. This played a crucial role in [\;BMS]\
;\, determining all perfect powers in the Fibonacci sequence. This "kit"
has also circulated in manuscript form since then.\n\nRecently\, Mignot
te and I have undertaken the task of making his "kit" manuscript ready f
or publication. This work is now complete and in this talk I discuss thi
s work. This work also includes several improvements to the initial manu
script.\n\nAs a demonstration of our improvements\, and to provide a ful
ly worked example that others can follow for application of this kit to
their own problems\, we rework the lower bounds for the linear form in &
#91\;BMS]\; used to show that there is no solution of y^P=F_n for n>1
2. We obtain an upper bound on p that is nearly 10 times smaller than th
e one obtained in [\;BMS]\;.\n\nPari/GP code for the application o
f the kit\, along with examples\, is also publicly available at https://
github.com/PV-314/lfl3-kit. It has already been used by researchers for
addressing several diophantine problems.\n\n[\;BHV]\; Y. Bilu\, G.
Hanrot and P. M. Voutier (with an appendix by M. Mignotte)\, Existence
of Primitive Divisors of Lucas and Lehmer Numbers\, Crelle's J. 539 (200
1)\, 75-122.\n\n[\;BMS]\; Y. Bugeaud\, M. Mignotte and S. Siksek\,
Classical and modular approaches to exponential Diophantine equations I
. Fibonacci and Lucas perfect powers\, Ann. Math. 163 (2006)\, 969-1018.
\n\nRoom: Frank Adams 1
STATUS:TENTATIVE
TRANSP:TRANSPARENT
CLASS:PUBLIC
LOCATION:Frank Adams 1\, Alan Turing Building\, Manchester
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