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Manchester Number Theory Seminar - Paul Voutier

Dates:21 February 2023
Times:15:00 - 16:00
What is it:Seminar
Organiser:Department of Mathematics
Who is it for:University staff, External researchers, Current University students
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  • Department of Mathematics

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  • In group "(Maths) Number theory "
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Speaker: Paul Voutier

Title: A kit for linear forms in three logarithms (joint work with Maurice Mignotte)

Abstract: Lower bounds for linear forms in logarithms are a powerful tool that have found application to many number theory problems. In fact, many problems can be reduced to linear forms in two or three logarithms. As a result, good lower bounds for such linear forms have resulted in the complete solution of several outstanding problems [BHV], [BMS].

In the early 2000s, Mignotte produced his "A kit on linear forms in three 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.

Recently, Mignotte and I have undertaken the task of making his "kit" manuscript ready for publication. This work is now complete and in this talk I discuss this work. This work also includes several improvements to the initial manuscript.

As a demonstration of our improvements, and to provide a fully 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 [BMS] used to show that there is no solution of y^P=F_n for n>12. We obtain an upper bound on p that is nearly 10 times smaller than the one obtained in [BMS].

Pari/GP code for the application of 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.

[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 (2001), 75-122.

[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.

Room: Frank Adams 1

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Frank Adams 1
Alan Turing Building
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Martin Orr

martin.orr@manchester.ac.uk

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