An experimental evidence on the validity of third Smarandache conjecture on primes

Felice Russo _\/tcr()J7 7ecil17()/o!!:1' 110/\' '.-1 re~~CIl7() (-1(1) IW(1' In this note v.e report the results regarding ,he check of the third Smarandache conjecture on primes [1 J.[2] for P ~ 2 and 2 ~ k ~ 10, In the range analysed the conjecture is true, n \loreover. according to experimental data obtained. it seems likely that the conjecture is true for all primes and for all positi\'e values ofk, J ntrod uction In [1] and [2] the follo\ving function has been defined: \vhere Pn is the n-th prime and k is a positive integer \loreover in the above mentioned papers the follov ... -ing conjecture has been fomlUlated by F, Smarandache 2 C(n,k)< k for k~2 This conjecture is the generalization of the Andrica conjecture (k=2) [3] that has not yet been proven This third Smarandache conjecture has been tested for P ~ 2 . 2 ~ k ~ 10 and in this n note the result of this search is reported The computer code has been written utilizing the L oasic software package Experimental Results In the follov,ing graph the Smarandache function for k=-1and n< I 000 is reported As we can see the value ofC(k.n) is modulated by the prime-s gap indicated by dn = Pn+l Pn We call this graph the Smarandache --comet"