On numbers written with a small number of digits in different numeration systems

The current year 2022 is written in the ternary numeration system as 2202220. We call a number n,m-lucky if its expansions in both n- and m-ary numeration systems contain only the numbers 0 and 1. Conjecture: the number of 10.3-lucky numbers is finite, but, for example, there are infinitely many 5.3-lucky numbers. In the talk, (non-rigorous) arguments in favor of this conjecture based on the Borel–Cantelli lemma will be given. A fast algorithm for finding such numbers will be described.