On the Properties of Self-Descriptive Numbers: Divisibility, Primality, and Structural Constraints

A self-descriptive number is an integer in a given base b whose digits describe the count of their
own occurrences within the number. This paper presents formal proofs for fundamental properties
of these intriguing numbers. We focus on bases b ≥ 4, as it is known that no such numbers exist for
b ∈ {1, 2, 3, 6}. While the fundamental properties of these numbers are established in recreational
mathematics literature, their proofs are often scattered or presented informally. This note aims to provide
a single, self-contained resource by presenting formal, elementary proofs for four foundational theorems
concerning self-descriptive numbers in bases b ≥ 4. We will demonstrate first that any such number is
necessarily a multiple of its base, and second, that no self-descriptive number can be a prime number.
Finally, we prove a strong structural constraint that holds for all bases: any self-descriptive number can
have at most four non-zero digits. We conclude by presenting, in an appendix, a rigorous proof of the
unique structural form for all bases b ≥ 7