regex to match irreducible fractions

Question

How can I match irreducible fractions with regex?

For example, 23/25, 3/4, 5/2, 100/101, etc.

First of all, I have no idea about the gcd-algorithm realization in

Answer 1

Nope it cannot be done. Like a good computer scientist I will ignore the specifics of the tool regex and assume you are asking if there is a regular expression. I do not have enough knowledge about regex's features to ensure it is restricted to regular expressions. That caveat aside, on with the show.

Rewording this we get:

Let L be the language {"a/b"| where a and b are natural numbers encoded in a radix r and a and b are coprime}. Is L regular?

Assume such a language is regular. Then there exists a DFA that can decide membership in L. Let N be the number of states of such a DFA. There are an infinite number of primes. As the number of primes is infinite, there are arbitrarily many primes greater than the largest number encodable in N digits in the radix r. (Note: The largest number is clearly r raised to the power of N. I am using this weird wording to show how to accommodate unary.) Select N+1 primes that are greater than this number. All of these numbers are encoded using at least N+1 digits (in the radix r). Enumerate these primes p₀ to pₙ. Let sᵢ be the state of the pᵢ is in immediately after reading the /. By the pigeon hole principle, there are N states and N+1 sᵢ states so there exists at least one pair of indexes (j,k) such that sⱼ = sₖ. So starting from the initial state of the DFA, inputs pₖ/ and pⱼ/ lead to the same state sⱼ (or sₖ) and pⱼ and pₖ are distinct primes.

L must accept all pairs of distinct primes p/q as they are coprime and reject all primes divided by themselves p/p as p is not coprime to p. Now the language accepts pⱼ = pₖ so there is a sequence of states from sⱼ using the string pₖ to an accepting state, call this sequence β. Let α be the sequence of states reading pₖ starting from the initial state. The sequence of states for the DFA starting at the initial state for the string pₖ/pₖ must be the same as α followed by β. This sequence starts in an initial state, goes to sₖ (by reading the input pₖ), and reaches an accepting state by reading pₖ. The DFA accepts pₖ/pₖ and pₖ/pₖ is in L. pₖ is not coprime to pₖ, and therefore pₖ/pₖ is not in L. Contradiction. Therefore the language L is irregular, or no regular expression exists.