I know that the language of prime numbers written in unary is not a regular language, but what about the prime numbers themselves? (When the alphabet is 0~9) By intuition, I think it's not a regular language, but how to prove it?
2026-04-01 18:32:16.1775068336
Is the set of prime numbers a regular language?
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The generating function of the number of words of a given length in a regular language is a rational function. This constrains the growth rate of these numbers. By the Prime Number Theorem, the number of primes of a given length (whether or not we admit leading zeroes) does not grow at a rate compatible with the generating function being rational.