The key ingredients in our proof of the special case of Cobham’s theorem were the Pumping Lemma and the following fact:
We can formally put them together into the following lemma:
Lemma 1. If is an infinite automatic set, then there exist distinct and integers and such that for all .
Proof. Say is -automatic. By the pumping lemma we can take for some with . Note that these are distinct as we don’t allow leading zeros. The conclusion now follows from .
Lemma 1 can be a powerful tool in proving that sets are not automatic, because it transforms a question from automata theory into the language of simple unary recurrences. For instance, we can use it to easily prove the following theorem of Minsky and Papert:
Theorem 1 (Minsky and Papert, 1966). The set of prime numbers is not automatic.
We are actually able to prove the following stronger result:
Theorem 2 (Schützenberger, 1968). An infinite set of prime numbers is not automatic.
Proof. Assume the contrary. Then by lemma 1 there exist distinct prime numbers and integers and such that for all . Then
for all . Now pick large enough so that . Then
by Fermat’s little theorem, a contradiction.
Note that lemma 1 implies that any set whose elements increase very rapidly cannot be automatic. To put it formally:
Theorem 3. Let such that as . Then the set is not automatic.
Proof. Assume the contrary. Then there exist integers such that for all , where and are fixed integers. Then as , a contradiction.
Corollary 1. The set of factorials is not automatic.
Proof. Take in theorem 3.
Corollary 2. The set of Fermat numbers is not automatic.
Proof. Take in theorem 3.
It is possible to use lemma 1 in other ways to show that sets are not automatic. For example:
Theorem 4. Let such that as , where is a real number such that for any . Then the set is not automatic.
Proof. Assume the contrary. Then by lemma 1 there exist distinct positive integers such that for all , where and are fixed integers. Now
for all . Therefore, as , it follows that for some . But this is impossible, as for any .
Corollary 3. The set of Fibonacci numbers is not automatic.
Proof. Let denote the -th Fibonacci number. Note that as , and that for all . So the result follows by theorem 4.
In the exact same manner we also get:
Corollary4. The set of Lucas numbers is not automatic.