An Erdős–Kac law for base-$b$ palindromes and for reversed primes

For every base $b\ge2$, the number of prime factors of the $\lambda$-digit base-$b$ palindromes, and of the base-$b$ reversals of the $\lambda$-digit primes, obeys an Erdős–Kac law: counted with or without multiplicity, it is asymptotically normal with centring $\log\log b^{\lambda}$ and scaling $\sqrt{\log\log b^{\lambda}}$. For reversed primes the law persists when the leading digit of the prime is prescribed. Erdős–Kac laws were already known for other digitally defined families — integers with a fixed digit sum, or with digits restricted to a fixed set — but for palindromes only the largest value of $\omega$ had been studied, with nothing known about the typical value, and for reversed primes the level of distribution the argument needs became available only in 2025.

New theorems, not a formalisation of previously known results. For every base $b\ge2$ the Erdős–Kac law is established for the $\lambda$-digit base-$b$ palindromes and for the base-$b$ reversals of the $\lambda$-digit primes, for $\omega$ and $\Omega$ and for $\omega_S,\Omega_S$ with any regular set $S$ of primes; with normal order $\log\log n$ on both families, and, for $\omega$, all moments of order up to $\tfrac12(\log\log b^{\lambda})^{1/3}$ uniformly in the order. For reversed primes it also holds with the leading digit prescribed. Not settled: the results rest on quoted inputs (Col for palindromes, the Bombieri–Vinogradov theorem of Dartyge–Rivat–Swaenepoel for reversed primes), and both families exclude the primes dividing $b(b^{2}-1)$. No rate of convergence is obtained. The question of Banks–Shparlinski on the *largest* value of $\omega$ on palindromes is untouched: the trivial bound $\ll\lambda/\log\lambda$ and their $\lambda^{o(1)}$ remain far apart.