A language over a one symbol alphabet requiring only O (log log n) space

A language over a one symbol alphabe t requiring only 0(log log n) spac e It is well known that the minimal growth function for th e tape complexity of Turing machines is log log n (21. In th e literature, one can find essentially one example of a language requiring only 0(log log n) space, namel y Lo ={ bin(1) $ bin(2) bin(3)-if. .. # bin(n)* ; n E IN } where bin(i) is the binary representation of the integer i. In this note we describe a language over a one symbo l alphabet having space complexity 0(log log n). Let L 1 = { an ; the smallest number q which does not divid e n is a power of two } For every natural number n let q(n) be the smallest numbe r which does not divide n. Lemma : 3c > 0 : q(n) S c log n Proof :. Let n be any natural number and let m = q(n). The n all primes less than m divide n and hence their product P divides them. By the prime number theorem of Gauss there ar e about m/ log m primes less than m and hence P is larger than