The Prime Numbers Hidden Symmetric Structure and its Relation to the Twin Prime Infinitude and an Improved Prime Number Theorem. Imre Mikoss Universidad Simón Bolívar, Dep. de Formación Integral y Ciencias Básicas, Valle de Sartenejas, Baruta, Caracas, Venezuela. email: [email protected] Due to the sieving process represented by a Secondary Sieving Map; during the generation of the prime numbers, geometric structures with definite symmetries are formed which become evident through their geometrical representations. The study of these structures allows the development of a constructive prime generating formula. This defines a mean prime density yielding a second order recursive and discrete prime producing formula and a second order differential equation whose solutions produce an improved Prime Number Theorem. Applying these results to twin prime pairs is possible to generate a “Twin Prime Number Theorem” and important conclusions about the infinitude of the twin primes. structure created by the construction of the primes, the Introduction reason why the twin primes should be infinite Most of the knowledge about the sequence of numbers becomes clear. This last is known as the Twin Prime 1 named primes is a set of unproved theorems and Conjecture , which is one of the unproved icons in the conjectures 1. The reason of this fact remains as elusive modern number theory. Using a mean prime density as the very proofs. The approach of this work proposes derived from the geometric construction, a discrete a new and heuristic way of treating these problems. As second order equation is obtained as well as a it is well known, sieving algorithms are the only continuous version of it, which is a second order non- efficient way to produce primes. This fact should be linear differential equation. This is a first taken as an indication that sieving is the natural way of approximation for a prime differential equation and is producing primes. A myth has been generated about used to demonstrate constructively and to improve the the sequence of primes, and many attempts have been best known approximation to the primes, known as the undertaken to find some properties which should be Prime Number Theorem. intrinsic to the sequence itself, despite its generating procedure 2. Most of them (perhaps all of them) are Sieving as a Recursive Map based on the famous Euler formula which relates the Usually recursive maps act on subsets of the real sequence of primes with the Zeta function 3. This numbers, although some of them are defined on formula is nothing else than an analytic representation geometrical objects 5. A recursive map which acts on of the sieving process 4. Through the construction of infinite and discrete sequences of numbers is proposed this formula a limit is taken, which eliminates the very here. This map called Secondary Sieving Map (SSM) heart of the process. Due to this limit, the relevance of is denoted with the Greek letter b. Given: the erased part has remained hidden from the scientific community through centuries. In the present work the structure of this hidden part is made evident through the iteration of a Secondary Sieving Map. Through the 2 However the number one does appear. This is a η={ ηηα , β , ηη χ , δ ,.... ∞} persisting characteristic in b’s successive applications An infinite and discrete sequence of natural on n: the number one always survives and the pivot numbers which satisfies: number used to generate the last iteration obviously disappears from it, because it was multiplied by one η< η < η < η < <∞ α β χ δ .... and extracted. In the second generation, the main Then b act on this sequence h in the following features of these sequences start to emerge: way: β2 ()()= ββ( ) = β 1 == 2 n n( nn()2) () 2,3 βη=− ηη* =− ηηη ⋅ {} = ( ) β 1,5,7,11,13,17,19,23,25,29,... {}{}1,7,13,19,25,...∪ 5,11,17,23,29,... = η=⋅ ηηηηηηηη ⋅ ⋅ ⋅∞ 1 * {βαβ , ββγβδ , , ... } ()()6⋅+n 1∪ 6 ⋅+ n 56 =⋅+ n 5 The minus sign means element extraction . The Where U means as usual the union of two sets temptation to factorize h should be avoided because and the last equation is a convenient way to write the the resulting equation is not the original; hb is a existence of two overlapped linear behaviors. Observe number whereas h is a set. In order to make the last that the set of pivot numbers (which record is kept in definition operational, h and h* should fulfill h û h*. n’s sub index) starts to form the set of prime numbers. The second element of the original sequence hb, is 2 1 Note also that the sequences n (2, 3) and n (2) are called the “pivot number”. The outcomes of applying qualitatively different: a splitting has occurred in the b is named generation. The natural numbers set first one. Instead of one linear function of n, there are (without the cero) * = n complies with all the two overlapped and simultaneous. In order to features required to be an argument of b. Applying b understand this segregation, the symmetric features of on n recursively is similar, but not equal, to the the SSM should be examined through a heuristic Eratosthenes sieve, because the last one is not applied geometrical representation of it, shown in the next on an infinite sequence and it was not conceived as an section. As a final sentence for this section, it should iterative mapping. The SSM applied to n, could be be noted that in the last decades some insight has been written in Mathematica ® as gained about a fractal structure of the primes 6, and it is “Nest[Complement[#,#[[2]]*#]&,Range[m1],m2]”, well known in dynamical systems that fractal where m1 is the size of the natural numbers subset on structures are produced by iterative mappings 5. The which it will be applied, and m2 is the desired SSM acting on n could be the basis for the fractal iterations number (it is impossible to act on a infinite structure of the primes. set with a computer). Acting once whit b on n produces the first generation, the set of odd numbers: Mirror Symmetry and Periodicity in Generations, as a Geometrical Image of a Multiple Linear β (n) = n1 ={ 1,3,5,7,...} =2.n+1 ()2 Representation Using this last notation, n 0 = n (generation The SSM can be represented geometrically using an cero). Observe that the first pivot number (the number infinite chain of curves (jumps) which connects the 2 which appears as a sub index in parentheses) used to sequence of numbers denoted with h* (see equations 1 generate de odd numbers, is not present in them. and 2). The non-touched numbers corresponds to the 3 Figure 1 | Generatio ns and Discrete Scale Invariance Graphical Representations . In Part A the first three generations geometric structure is shown through an analogy between extracted and touched numbers. The periodicity in each generation and their mirror symmetries are evident in the structures of the touching curves. In Part B the DSI of the Second Generation is shown. first generation. Further iterative applications of b can mentioned in the past section is caused by the be made on the same graph simply overlapping the incommensurability of the first pivot number (2) and geometrical representations of the corresponding h*. the second (3) which sets the first untouched numbers For example in Figure 1, the representations of three (1 and 5) to lie symmetrically around the number generations are seen in separate lines. The touched three. Then, due to the six-fold periodicity, the two numbers under b’s action, are just touched once linear behaviours represented in equation (3) are whereas in the graphical representation multiple produced. Note that mirror symmetry around the touching is allowed. In this way the geometrical numbers 3, 9, 15… as well as around the numbers 6, features are better observed. From these drawings can 12, 18… starts to emerge. This symmetry will become be inferred that due to the successive applications of more evident in the third generation and afterwards: the SSM, periodic structures form spontaneously. The β 3(n) = n 3 ={1,7,11,13,17,19, 23, 29,31...} = period of a particular generation is given by the ()2,3,5 1 31 61 91 121 151 181 ... 1 multiplication of all the previous pivot numbers (the 7 37 67 97 127 157 187 ... 7 formal demonstration will be published elsewhere). 11 41 71 101 131 161 191 ... 11 13 43 73 103 133 163 193 ... 13 Actually these structures are periodic both in the =30 ⋅n + 17 47 77 107 137 167 197 ... 17 generations but also in the superposition of the h* 19 49 79 109 139 169 199 ... 19 produced and extracted during each generation. In fact 23 53 83 113 143 173 203 ... 23 this last superposition is the most notorious in the 29 59 89 119 149 179 209 ... 29 geometrical representations from Figure 1, Part A. As it can be observed in the same figure, the splitting 4 This last equation is again a condensed and through its generation will certainly forecast all the convenient way to represent the third generation. In primes between itself and its square, this interval is Figure 1 the corresponding representation shows a called PCI. The PCI is depleted from the pivot number new period (30 = 2.3.5) and a new mirror symmetry multiples and the other numbers contained within have around the multiples of 30 and their halves. In order to their squares, cubes, etc certainly in the outside of this advance in a description of the subject, some interval.