On periods of the Tribonacci sequence modulo primes

In this talk, we study the \(p\)-adic zeros of the Tribonacci sequence \(T_n\), for a prime \(p\). In particular, we complete the analysis that was done in [1] by investigating the case of \(p = 11\). In addition, we investigate the primes \(p\) for which at the first two values \(T_n\) and \(T_{n−1}\) which are both divisible by \(p\), both of these are actually divisible by \(p^2\) , proving under the assumption of a generalisation of the \(abc\) conjecture that there are infinitely many \(p\) which are not such.

This talk is based on joint work with Liam Baker and Florian Luca.

References:

[1] Yuri Bilu et al. “On the p-adic zeros of the tribonacci sequence”. In: Mathematics of Computation 93.347 (2024), pp. 1333–1353. issn: 0025-5718,1088-6842. doi: 10.1090/mcom/3893.