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