OEIS A000350: Fibonacci numbers ending with their index in a base

We dive into A000350, the Fibonacci-ending-in-M problem across bases. In base 10, there are many nontrivial M, suggesting rich, infinite variation. In binary, the situation collapses: only M = 0, 1, 5 work—a result finally proven by Max Alexei after extensive computation (up to M < 2^25). We trace the history back to mid-1960s Fibonacci Quarterly work and pose questions about other bases like base 3 or base 8, illustrating how changing the base reshapes the underlying number-theoretic landscape.

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