Erdős–Kac laws for palindromes and reversed primes
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Do prime-factor counts of base-b palindromes and of reversed base-b primes obey an Erdős–Kac normal law for every b≥2? Both distinct prime factors and factors counted with multiplicity are considered.
Why it matters and what counts as progress
Why it matters: Normal laws describe typical factorization in digit-restricted families, beyond bounds on unusually large prime factors.
What counts as progress: A normal-limit theorem for each stated family and counting convention, including any leading-digit restrictions and exceptional primes in the argument.
How the problem developed
- 2005 · Partial result [1]
William D. Banks and Igor E. Shparlinski Studied prime divisors of palindromes and obtained large-factor results; these extremal results did not give an Erdős–Kac law for typical palindromes.
- 2009 · Partial result [2]
Stéphane Col Established a distribution result for palindromes in arithmetic progressions, an input to later prime-factor statistics rather than a normal-law result.
- 25 Jun 2025 · Partial result [3]
Cécile Dartyge, Joël Rivat and Cathy Swaenepoel Proved a Bombieri–Vinogradov type theorem for reversed primes and showed many have a reverse with boundedly many prime factors; this supplies a distribution input, not the Erdős–Kac law.
The pseudonymous author with Claude models Reported Erdős–Kac laws, with or without multiplicity, for base-b palindromes and reversed primes for every b ≥ 2, including reversed primes with fixed leading digit. The author credits Claude models with the modified moment criterion and draft and Lean development.
AI contribution
An Erdős–Kac law for palindromes and for reversed primes
The archived preprint proves Erdős–Kac normal laws for the number of prime factors, with or without multiplicity, of base-b palindromes and reversed base-b primes for every b≥2. The reversed-prime theorem also allows a fixed leading digit.
Sources
- Banks and Shparlinski, Prime divisors of palindromes
- Stéphane Col, Palindromes dans les progressions arithmétiques
- Dartyge, Rivat and Swaenepoel, Prime numbers with an almost prime reverse
- Archived Erdős–Kac manuscript
- AIProofWatch, An Erdős–Kac law for palindromes and for reversed primes
- Archived preprint DOI
- Lean theorem map
- Lean verification and axiom report
Website timeline
Publication dates and editorial changes.
AI-assisted criterion yields Erdős–Kac laws for palindromes and reversed primes