

An experimental evidence on the validity of third Smarandache conjecture. Felice Russo Micron Technology Italy Avezzano (Aq) – Italy
Abstract In this note we report the results regarding the check of the third Smarandache conjecture on primes [1],[2] for and . In the range analysed the conjecture is true. Moreover, according to experimental data obtained, it seems likely that the conjecture is true for all primes and for all positive values of k..
Introduction In [1] and [2] the following function has been defined: where is the nth prime and k is a positive integer. Moreover in the above mentioned papers the following conjecture has been formulated by F. Smarandache: for This conjecture is the generalization of the Andrica conjecture (k=2) [3] that has not yet been proven. The Smarandache conjecture has been tested for , and in this note the result of this search is reported. The computer code has been written utilizing the Ubasic software package.
Experimental Results In In the following graph the Smarandache function for k=4 and n<1000 is reported. As we can see the value of C(k,n) is modulated by the prime’s gap indicated by . We call this graph the Smarandache "comet".
In the following table, instead, we report:
According to previous data the Smarandache conjecture is verified in the range of k and analysed due to the fact that is always positive. Moreover since the Smarandache function falls asymptotically as n increases it is likely that the estimated maximum is valid also for . We can also analyse the behaviour of difference versus the k parameter that in the following graph is showed with white dots. We have estimated an interpolating function: for k>2 with a very good value (see the continuous curve). This result reinforces the validity of Smarandache conjecture since:
New Question According to previous experimental data can we reformulate the Smarandache conjecture with a more tight limit as showed below?
where and is the Smarandache constant [4],[1]. References [1] See http://www.gallup.unm.edu/~smarandache/ConjPrim.txt [2] Smarandache, Florentin, "Conjectures which Generalize Andrica's Conjecture", Arizona State University, Hayden Library, Special Collections, Tempe, AZ, USA. [3] see: "Andrica’s conjecture" in http://www.treasuretroves.com/math/ [4] N. Sloane, Seq. A038458 ("Smarandache Constant" = .5671481302020177146468468755...) in <An OnLine Version of the Encyclopedia of Integer Sequences>, http://www.research.att.com/cgibin/access.cgi/as/njas/sequences/ eishis.cgi. 