The Skolem landscape

The celebrated Skolem-Mahler-Lech theorem states that the set of zeros of a linear recurrence sequence is the union of a finite set and finitely many arithmetic progressions. The corresponding computational question, the Skolem problem, asks to determine whether a given linear recurrence sequence has a zero term. Although the Skolem-Mahler-Lech theorem is almost 90 years old, decidability of the Skolem problem remains open. Prof Florian Luca starts his inaugural lecture by presenting classical results on this problem, and then discusses two recent contributions that he and his co-authors have made in this regard. The first is an algorithm to solve the Skolem problem for simple linear recurrence sequences (those with simple characteristic roots). Whenever the algorithm terminates, it produces a standalone certificate that its output is correct – a set of zeros together with a collection of witnesses that no further zeros exist. The algorithm always terminates assuming two classical number-theoretic conjectures. When, in a preliminary experiment, the algorithm was implemented within the SKOLEM tool, the results pointed to the practical applicability of the method. The second contribution is the notion of a universal Skolem set, which is a subset of the positive integers on which the Skolem is decidable, regardless of linear recurrence. Prof Luca offers two examples of such sets, one of which is of positive density (containing a positive proportion of all the positive integers) and perhaps even of density 1.

WATCH THE INAUGURAL LECTURE HERE

Short biography

Florian Luca is a professor of Mathematics at Stellenbosch University (SU). Before joining SU in 2024, he was a distinguished research professor at Wits University. He has held various appointments abroad, including at Syracuse University (United States), Bielefeld University (Germany), the Czech Academy of Sciences, and the National Autonomous University of Mexico. He holds a PhD in Mathematics from the University of Alaska Fairbanks (1996).

Prof Luca specialises in number theory with an emphasis on Diophantine equations, linear recurrences and the distribution of values of arithmetic functions. He has made notable contributions, in particular to proving that irrational automatic numbers are transcendental, and proving a conjecture of Erdos on the intersection of Euler’s totient function and the sum-of-divisors function. He has produced over 800 papers with more than 350 co-authors. He currently serves as a principal investigator in the research project DynAMiCs (2025-2031), which is funded by the European Research Council. Conducted alongside colleagues from the Max Planck Society and the French National Centre for Scientific Research (CNRS), the project seeks to advance research that combines Mathematics and Computer Science to make progress on difficult open problems in both these areas.

In recognition of his work, Prof Luca has received various international awards, such as an Alexander von Humboldt fellowship (Germany) in 1998, a Guggenheim fellowship (United States) in 2005, and a CNRS ambassador fellowship (France) in 2024. On the local front, he is a member of the African Academy of Science (AAS) as well as the Academy of Science of South Africa (ASSAf), and has been A-rated twice (2016 and 2022) by the National Research Foundation.