Abstract
The rich history of prime numbers includes great names such as Euclid, who first analytically studied the prime numbers and proved that there is an infinite number of them, Euler, who introduced the function , Gauss, who estimated the rate at which prime numbers increase, and Riemann, who extended to the complex plane z and conjectured that all nontrivial zeros are in the axis. The nonadditive entropy , where BG stands for Boltzmann-Gibbs) on which nonextensive statistical mechanics is based, involves the function . It is already known that this function paves the way for the emergence of a q-generalized algebra, using q-numbers defined as , which recover the number x for . The q-prime numbers are then defined as the q-natural numbers , where n is a prime number We show that, for any value of q, infinitely many q-prime numbers exist; for they diverge for increasing prime number, whereas they converge for ; the standard prime numbers are recovered for . For , we generalize the function as follows: (). We show that this function appears to diverge at , . Also, we alternatively define, for , and , which, for , generically satisfy , in variance with the case, where of course .
Highlights
Extending the realm of the Boltzmann-Gibbs-von Neumann-Shannon entropic functional, many measures of uncertainty have been proposed to handle complex systems and, complexity
The qprime numbers are defined as the q-natural numbers hniq ≡ elnq n (n = 1, 2, 3, . . . ), where n is a prime number p = 2, 3, 5, 7, . . . We show that, for any value of q, infinitely many q-prime numbers exist; for q ≤ 1 they diverge for increasing prime number, whereas they converge for q > 1; the standard prime numbers are recovered for q = 1
We focus on the q-generalized numbers hniq, n ∈ N, given by Equation (4), to see how far we can conserve the essential concept of prime numbers in the set of q-numbers
Summary
Extending the realm of the Boltzmann-Gibbs-von Neumann-Shannon entropic functional, many measures of uncertainty have been proposed to handle complex systems and, complexity. There are different ways of defining generalized q-numbers connected with the pair of inverse (q-logarithm, q-exponential) functions, namely h x iq q hxi. Generalized arithmetic operations follow from each of the q-numbers and, consistently, there are various possibilities. The present paper will only explore one possibility for q-numbers, namely our Equation (4), equivalent to the iel-number Equation (11a) of [8]. For this choice, two algebras will be focused on here, namely, and h x i q #qhyiq ≡ h x ◦ y i q (8). The other algebra, corresponding to Equation (9), is presented, and it corresponds to the oel-arithmetics addressed in Section III.D of Ref. The other algebra, corresponding to Equation (9), is presented in Section 5, and it corresponds to the oel-arithmetics addressed in Section III.D of Ref. [8], where {q} # is here noted #q
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