Abstract :
The one-dimensional random walk (RW), where steps up and down are performed according to the occurrence of special primes, is defined. Some quantities characterizing RW are investigated. The mean fluctuation function F(l) displays perfect power-law dependence F(l) l1/2 indicating that the defined RW is not correlated. The number of returns of this special RW to the origin is investigated. It turns out that this single, very special, realization of RW is a typical one in the sense that the usual characteristics used to measure RW, take values close to the ones averaged over all random walks. This fact suggests that random numbers of good quality could be obtained by means of RW on prime numbers. The fractal structure on the subset of primes is also found
