7/23/2019 AToTHePIsNotRegular(EquivalenceClasses)r

http://slidepdf.com/reader/full/atothepisnotregularequivalenceclassesr 1/1

Consider Σ = {a} and L = {a p : p is a positive prime integer}.

If m is prime and n is not prime, then am and an are in different equivalence classes.

What about and !" #he string a distinguishes them.

! and $" %ot distinguished b& a'  for '=(, ), , !. *ut a+ does distinguish them.

Is it the case that a p and aq p-q are al/a&s in different equivalence classes"

I claim that for some '0(, the string a'q1p distinguishes a p and aq.

Lemma.  #here is no infinite arithmetic sequence of primes.

Proof of theorem using the lemma

We 'no/ that p 2 )q1p is prime. Let ' be the smallest positive integer such that p 2 '2)q1p is not

 prime.

#hen a pa p2'q1p is in L, and aqa p2'q1p  = a p2q1pa p2'q1p = a p2'2)q1pis not in L

Proof of lemma. Let n be the difference bet/een t/o consecutive terms in an arithmetic sequence.

Consider n2)3 2 ). #he ne4t n consecutive integers are composite.

54amples: n2)3 2 is divisible b& . n2)3 2 ! is divisible b& ! 6

n2)3 2 n is divisible b& n n2)3 2 n2) is divisible b& n.

7o none of these n consecutive numbers is prime.

Final conclusion. #he language L has infinitel& man& equivalence classes.