Happy Numbers Across Bases: A Deep Dive

We explore the simple rule of summing squares of digits and how iterating it reveals happy numbers in base 10 and beyond. Formalizing with f_{2,b}(n), we trace examples like 13 and 4, examine cycles, and discuss arithmetic dynamics across bases. We'll see why there are infinitely many happy numbers in any base, how density behaves without settling to a single value, and what makes bases like 2 and 4 “happy” and others like 6 exhibit cycles. We’ll also touch on open questions about which bases are completely happy and the nature of the resulting cycles.

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