bibliography on this topic). From a practical point of view, many tricks can be used to find all primes less than 10 12 in a fast way, as explained for example in [1]. Clearly the enumeration of all the primes less than x cannot have a lower cost than (x). Besides the computation of (x), the number of primes less or equal to x, does not need to find all the primes less than x. This fact is set up by the formula of Legendre, which uses the prime numbers less or equal to p x. Next, the works of Meissel and Lehmer provides more subtle formulae, which reduce the amount of computation. As an example Meissel computed the value of (10 8 ). Nevertheless, these methods all have a cost of O(x 1+" ). Lagarias, Miller, and Odlyzko gave a method which for the fi