A Solution to the Famous “Twin’s Problem”

In the following pages I will try to give a solution to this very known unsolved problem of theory of numbers. The solution is given here with an important analysis of the proof of formula (4.18), with the introduction of special intervals between square of prime numbers that I call silver intervals M δ . And I make introduction of another also new mathematic phenomenon of logical proposition “In mathematics nothing happens without reason” for which I use the ancient Greek term “catholic information”. From the theorem of prime numbers we know that the expected multitude of prime numbers in an interval [ ] , x x dx + is given by formula  considering that interval as a continuous distribution of real numbers that represents an elementary natural numbers interval. From that we find that in the elementary interval ) νν+ around of a natural number ν we easily get = the ν  that has the ν to be a prime number. From the last formula one can see that the ν > of formula (4.18) is absolutely in agreement with the above theorem of prime numbers. But the benefit of the (4.18) is that this formula enables correct calculations in set N on finding the multitude of twin prime numbers, in contrary of the above logarithmic relation which is an approximation and must tend to be correct as ν tends to infinity. Using the relationship (4.18) we calculate here the multitude of twins in N, concluding that this multitude tends to infinite. But for the validity of the computation, the distribution of the primes in a random silver interval M δ is examined, proving on the basis of catholic information that the density of primes in the same random silver interval M δ is statistically constant. Below, in introduction, we will define this concept of “catholic information” stems of “information theory” [1] and it is defined to use only general forms in set N, because these represent the set N and not finite parts of it. This concept must be correlated to

δ . And I make introduction of another also new mathematic phenomenon of logical proposition "In mathematics nothing happens without reason" for which I use the ancient Greek term "catholic information". From the theorem of prime numbers we know that the expected multitude of prime numbers in an  (4.18) is absolutely in agreement with the above theorem of prime numbers. But the benefit of the (4.18) is that this formula enables correct calculations in set N on finding the multitude of twin prime numbers, in contrary of the above logarithmic relation which is an approximation and must tend to be correct as ν tends to infinity. Using the relationship (4.18) we calculate here the multitude of twins in N, concluding that this multitude tends to infinite.
But for the validity of the computation, the distribution of the primes in a random silver interval M

Introduction
The symbol q ν (with N ν ∈ ) from now on will symbolize the prime numbers.
There are the definitions of the endless sequence of the prime numbers that will be symbolized as follows: 0 Let a random natural number a ν ν = , N ν ∈ and let two more prime numbers q α , q β that are not equal to each other, q q α β ≠ . At first, it will be shown the independency of the fact that "the random Natural number a ν , can be divided by the prime number q α " from the fact that "the random Natural number a ν can be divided by the prime number q β ". Let, also, without impairment of the generality of this proof, that 7 q α = and 5 q β = . Obviously, per 35 successive Natural numbers the 5 Natural numbers will be multiple of 7 and the 7 Natural numbers will be multiples of 5 and only one Natural number will be multiple of both 7 and 5. So, when selecting an Integer number a ν from the infinite multitude of Natural numbers, the information of the fact that " a ν is multiple of 7" does not interfere with the probability of the fact that " a ν is multiple of 5" because every five multiples of 7 there will be only one that can be divided once again by the prime number 5, regardless of the information of the first fact. The Natural number a ν will indeed belong to a group of thirty-five, if the multitude of Natural number N is divided into groups of thirty-five successive Natural numbers. Therefore, if the first fact, " a ν is multiple of 7", is valid then a ν will belong to the group of 5 (of a group of 35) that are multiples of 7. This group of 5, however will include only one multiple of 5, therefore a ν will again have 1/5 probability of being multiple of 5, regardless of the information of the first fact that "it is multiple of 7". The proof is obviously generalized with the same methodology for any of the prime number q α , q β , not equal to each other.
It should be underlined that the Natural number 0 is divided by every Natural number, even when selecting a random integer a ν , which will obviously have a probability equal to 1 k q of being divided by the prime number k q . Here, the probability has the meaning of the appearance frequency of a subset of Natural numbers, so when stating that fact Γ is independent of the probability-frequency that is referring to the subset of these Natural numbers, defined; based on an activity-criterion of their selection, it is meant that fact Γ is independent from the activity-criterion of their selection.
Opposite to that now, the information "a Natural number a ν is divided by one non-prime number (compound) i.e. 18" immediately gives the information that a ν will be divided by all the prime numbers that divide 18, therefore by 2 Every random element selection from a given set A will be called Catholic Selection (CS), a term from ancient Greek language. In addition, as Catholic information will be defined a set K of catholic (logical) propositions that will be valid for infinity CS elements from different appropriate subsets (of finite multitude) of a set A. For example, a set K consisting of finite multitude of relations (written in general form) among infinite elements of another set A. An example of a logical proposition (or simply a proposition) that is a Catholic Proposition (CP), due to the fact that it is valid for infinite elements in N, specifically for infinite pairs of multiples of two prime numbers each time, is the PDI. The set of all the CP, meaning the propositions of catholic cardinality in N, that can be proved using PDI will now defined as catholic information of PDI for N. Owing that in mathematics nothing happens without a reason, it is concluded that if an algorithm of creation of a set A with infinite multitude is proven that does not create a property (proposition) P that will be catholically valid in A, therefore not implied by this algorithm (a set of finite multitude propositions) that the proposition P (e.g. a non-random statistical distribution) is valid in A, then P will not be valid in A. This last sentence will be named Proposition of Catholic Information.

The Fundamental Principles of This Research
They are: 1) The introduction of "catholic information" that we introduced above.
2) The extraction of all possible catholic or general relationships for which we define to must being them valid in all parts of set of natural numbers N and not only for special parts [2]. So the catholic formulas must use alphanumerical (by general expression) symbols for their catholic variables.
3) The definition of two kinds of intervals which we here call silvers and darks respectively.
4) The statistical calculation of catholic multitude of twin prime numbers in set N that is a calculation until infinity.
compact calculation using the appearance frequencies of twins in N, and second by using the dark intervals of N, which are increasing their sizes by a monster rate and however they are have infinite multitude. In this last process we initially proof that if some intervals not includes twin primes then this hypothesis drives to the existence of one twin prime on every top of these intervals. Thus we again arrive on the same conclusion. In pages before the relation (4.20), we examine the stability of frequency of prime numbers appearance in a random "silver interval", which is, a condition useful of validity of statistical calculations bellow.

Conclusions
Our conclusions from the below are: 1) The hypothesis of twin prime numbers is correct. 2) Maybe the concept of "catholic information" can be used as well in other Mathematical investigations. This concept is the other expression of fundamental proposition that "In mathematics nothing happens without reason".
This concept of "catholic information" maybe could be connected by Riemann hypothesis [3] [4].

Twin Pairs
Here will be studied the twin pair problem. Let that an "honest" dice (in the shape of a normal hexagon) is thrown three consecutive times and the three consecutive positions are noted respectively A/B/Γ. Which twin pairs (meaning repetitions) of a particular number, i.e. of 5, are expected?
Answer [5]: According to the sample space 6(6)6 = 216 facts there will be five cases of the form 5/5/C, where Γ is one of the results {1, 2, 3, 4, 6}, which has multitude of five. Similarly, there are five cases of the form Α/5/5, where Α is one of the results {1, 2, 3, 4, 6}, and only one is the 5/5/5. Due to the 10 first having 1 twin pair of 5, meaning one boundary "/" for the 5, and the last (one) case having 2 twin pairs, there will be altogether 10(1) + 1(2) = 12 total twin pairs in the 216 cases, and therefore the probability of twin pairs in A, B, Γ facts, (which is "the 5" in each one of the ordered throw) will be 12/216 = 1/18. On the other aspect of the counting method based on the probability 1 6 p = to get number 5 in a throw, there will be probability ( )( ) 2 1 6 1 6 1 36 p = = for the twin pair of A, B and similarly 1/36 for the twin pair of B, Γ, so expected probability 1/36 + 1/36 = 2/36 = 1/18, which shows that in this example the two non-independent and at the same time non-incompatible facts X A B = ∩ and Y B = ∩ Γ will be counting their respective percentages without considering their dependency and compatibility. The condition, however, for the proper counting is the independency of A, B, Γ which is true. Therefore, the expected multitude of twin pairs of 5 will be 216•(1/18) = 12 cases of twin pairs, as found above. Generalizing the above problem for N successive throws of a (fair) dice the expected percentage of the twin pairs of the number 5, of the "fair" dice, for the N − 1 multitude of the Advances in Pure Mathematics boundaries "/" of the facts of the throws 1 2 / / / N A A A  would be: One could try to prove this last relation in the case of a/b/c/d using the first method with the sample space. However, in this case the counting of the probability P 0 will be completely different in order for at least one of the above facts X and Y to occur. This probability will be counted as follows: The probability ( ) 0 | P Y X above is 1/6, because when fact X occurred the information that the second pair has already given 5 is provided, so the ( ) 0 | P Y X will correspond only to the probability "the third dice will be again 5", and it is obviously 1/6. In the sample space of the 216 facts there will indeed be 5 + 5 + 1 = 11 of these cases (and not 12 as before), since 5 cases will be of the form 5/5/Γ, the other 5 of the form A/5/5 and only 1 will be 5/5/5. The reader perceives that the differentiator is the key phrase in the above sentence: at least one.
Coming to an end, by proving the independency of the events; the "divisibility of the random natural number a ν by the random prime number q α (of its sub-sequence)" from the "divisibility of the random natural number a µ by the random prime number q β (of its sub-sequence)" the definition of these two events independency will be repeated: Any two events Γ 1 and Γ 2 will be considered independent from each other in a set (their range) A, "if and only if the frequency-probability of the elements in A where Γ 1 appears, is the same as the frequency-probability of the elements in A where Γ 1 and Γ 2 appear together (once each)" and additionally the last sentence ("…") is valid if C 1 and Γ 2 are interchanged in it.  Because if this last statement were to be   true, then according to the aforementioned facts there would be a divisor smaller than a ν , which would either be prime or it would be analyzed in product of prime numbers that are for sure smaller than a ν . The conclusion drawn is that in the case where a ν , does not have as a divisor a prime number smaller than a ν , then a ν is the prime number. Therefore, the prime numbers that define as possible divisors of a ν being prime, are only the prime numbers that are all smaller than its square root, which is the sub-sequence of prime numbers of a ν , that was defined above.

Specification of the Indefinite Frequency-Probability Appearance of Prime Numbers
The probability P ν , that a ν ν = is not divided by any of the elements of the sub-sequence of 2,3,5, 7, , M q ν  (defining 0 1 2 3 1, 2, 3, 5, q q q q = = = =  ) will be equal to the products of the probabilities not to be divided by 2,3,5, 7, , M q ν  .
These probabilities will respectively be 1 1 that state respectively that the natural number a ν ν = is divided by the prime numbers 2,3,5, 7, , M q ν  of its sub-sequence, which according to PDI that was previously proven, per two events that are independent from each other. It is obvious that 1/2 is the probability of the natural number a ν , to be divided by 2, that is to be an even number with complimentary probability the 1 − (1/2) not to be divided by 2. Similarly, 1/3 is the probability of a ν to be divided by 3, while 1 − (1/3) is the complimentary probability to not be divided by 3 and so on for every term of the sub-sequence. The 1 2 3 , , , , M A A A A ν  , however are not every two exclusive events from each other, owning to the fact that the divisibility of the natural number a ν ν = by a number of its sub-sequence do not exclude its ability to be divided by another term of that sub-sequence. For example, the natural number 30 30 a = has as a sub-sequence of prime numbers 2, 3, 5 and the fact that it can be divided by another of these three terms. It is indeed divided by the term 3. The probability P ν of the following Equation (2.1) is a unique enumerate of prime numbers, but (initially) in not-well-defined intervals. The following definition is derived from the available information of the production of infinite element of set N, provided that according to the definition of Shannon the probability P ν is another way of expressing information. Based on the fact that the events 1 2 3 , , , , M A A A A ν  are every two independent from each other, one concludes that the probability-frequency of appearance of all the events above will simply be the product of all their individual probabilities therefore one will obtain the relation.
It will, however be proven and in another way the relation (1.2) [5]. Let P ν the probability that the natural number a ν is divided with at least one term of the sub-sequence of its prime numbers. Then the probability not to be divided by any of its terms will obviously be complimentary of the probability: and because the facts 1 2 3 , , , , are as previously mentioned per two independent from each other, the relation above becomes The second part of the equation in the last relation is due to the independency of the fact So, the probability P ν results to expression ( ) In the above sums the indicators 1 2 3 , , , j j j  are as known, per two different from each other and obviously and also ( ) ( )( ) ( ) The relations (2.3), (2.5), (2.6) result in ( ) 1  Q P ν δ ν ν = ∑ . The exact counting as shown below, will be done in appropriate intervals, that have already been named silver intervals M δ , and with the use of an unknown probability P ν , that will be proven to be greater than a useful expression, that will be related to (2.1). From the above it is becoming clear that all the natural number that have the same sub-sequence of prime numbers should constitute an interval such as M δ , i.e. the intervals: ( ) ) ) ) noticed that the first of the above silver intervals, that is 0 δ , has as a sub-sequence the empty set and includes two prime numbers which are 2 and 3, the second 1 δ has as its sub-sequence the unit-set with 2 as an element and includes two prime numbers, 5 and 7, while the third one includes five prime numbers, the fourth includes sixteen prime numbers and so on. Furthermore, the enumerators-probabilities that were mentioned, P ν and P ν , will have constant value in every specific silver interval, which will be explained in details, and be proven in Section 4. In this section it will be defined that these values will be dependent, according to relation (2.1), only on the sub-sequence of prime number, which is the same for all natural numbers and only of the specific silver interval: with κ function of ν.
Now certainly the definition of silver intervals is justified ν an arbitrarily large natural number, it is shown that the countable multitude of prime numbers, whilst at the beginning coincides with the real, it becomes more and more larger than that of the real multitude of prime numbers, as 2 ν is increased. The reason why this is happening will be explained below and will be proven that the new precise probability ( ) p P ν ν = , that was mentioned before will tally the precise multitude of prime numbers: a very useful inequality, which will be named fundamental inequality of the silver intervals.
It will also be proven true that for the probability of the relation (2.1): In the beautiful book "the secret life of numbers" professor of Mathematics Andrew Hodges mentions that one of the smartest tricks in the history of mathematics is the Euler transformation below: The second part of 2.10 is the known harmonic sequence that as known is inexhaustible and corresponds to Riemann's function: The proof of (2.10) results directly from the general form of writing the natural number: That was mentioned in the beginning of Section 2. Executing retrospectively the multiplication of the 1 st part it will indeed lead to the 2 nd part due to the appearance of all the combinations of (2.11) in the denominators, and so all the integers positive numbers etc.
The relation (2.10) is known from the time of Gauss, that leads directly to the conclusion found by Euclid thousands years ago, which is that the prime numbers are infinite. If they were not then the first part of (2.10) would be a product of finite multitude of derivatives, where each one of them would converge and therefore this product would not deviate from infinity. This however, is absurd, since the second part would also not deviate, which is indeed deviating, because it is the harmonic sequence that was previously mentioned. However, the author's shorter proof can be given here: "the relation (2.11) includes exponents that are natural integer numbers and therefore each one of them is developed again in the same way will be developed again in the same way and so on. It is therefore obvious, that if the prime numbers had finite multitude in N, then the combinations for the representation the natural numbers a ν would be depleted, since these combinations would not obviously have the advantage of infinite different per-two mathematical (tree-like) representations. Hence, in that case the infinite natural numbers would not be represented by the relation (2.11) which is absurd".
One more not so well known relation of the bibliography (that is also mentioned in Section 6 of Andrew's Hodges book) for a random prime number q ν , as symbolized here, is: It is noted that (2.12) could be proven easily from the known Taylor formulation [3]: raised in all the powers, to infinity will appear. Hence, according to the relation (which was reported in the beginning of this Section 2) the result will be all the natural numbers, therefore function ( ) 1 ζ . Combining this fact with the one from (2.10) (2.12) and also with (2.1) the following known relation is concluded:

The Tracker of Infinity (Eratosthenes Sieve)
One should think about the endless axis of positive natural numbers a ν , that starts from 0 and contains, in equal distances, the natural numbers 1, 2, 3, 4, 5, 6, ral number a ν send on your right, to the abyssal infinity, a message to the light blue observers-natural numbers, which are integer multiples of a ν , that says change your colour to black. The a ν remains light blue and is registered in your log book". What will happen? Simply. In the route 2 3 → all the even natural numbers will be black to infinity except of course for number 2. These will be called second-multiple (2-multiples) not including number 2. In the route 3 4 → there will be black numbers except from the second-multiples and all the multiples of 3 to infinity except for the natural number 3. These multiples of 3 not including the initial number 3 will be called third-multiples (3-multiples) numbers. When the tracker reaches number 4, however, finds it black and does not send a message for colour changing to the observers-natural numbers.  , the tracker will encounter, marked in black, all the multiples of 2 and 3. That means that the multiples of the subsequence of the prime numbers in the silver interval that the tracker crosses each time, will be marked black.
The prime numbers that are integer multiples of the random prime number s q , will be called s q -multiples. Additionally, in a random silver interval the natural numbers will be called: Greek word "τελευταίος" that means last). The interval  that includes only natural numbers will be called band of s q -multiples (prime-multiples) of the silver interval M δ . Furthermore, The "equal to 0" in the first relation (3.3) represents the case where s M q q = .
For example:

The Silver Intervals, the Fundamental Inequality and a First Solution
Summing up, the silver interval is defined as M δ using the relation Between the first (1)   b M band will be called, as stated before its geometrical length. At last the relation (3.2) was proven: It should be clarified that all the composites of M δ are necessarily s q -multiples of its subsequence, as it was proven in Section 2.
In the below Figure 1 we see that the band as the (4.5) implies above, however that was done die to the simple supervision and obviously it will not interfere with the proving methodology that will be shown below. (A composite natural number is the one that is not prime). In general all the prime-multiples numbers of a random prime number s q (meaning all its integer multiples, that were named s q -multiple numbers) function as erasers in the list of prim number candidates, since they erase the possibility of being the natural numbers with which the prime numbers coincide. Using this interpretation they will be named s q -multiples erasers. These are the ones that the tracker changes their color from light blue to black (erasure) as soon as the tracker meets the first s q number on the unending travel of the axis a ν ν = of natural numbers.
In Figure 1 are shown the two boarders  ( ) is based on the assumption that the density of j q -multiples erasers in the silver interval M δ , where 1, 2,3, , j M =  is: Thus, the active multitude of j q -multiples erasers in M δ , with "length" M d will be: Hence, the non-erased natural numbers in M δ from the j q -multiples will have as active multitude δ and are of multitude n + 1 (with n = 3 in Figure 1). Hence, in general it will be true: However due to the relation (4.8), (4.9) and based on the help of Figure 1 in the general form it would be The remainders between the silver interval M Combining the last relation with the expression of s K from above, it is given: And based on (4.11) it will be true This relation (4.13) shows how the expected multitude of prime numbers in the M δ interval will have to be as a whole greater than the true multitude, since the active multitude of the erasers is generally smaller that their true multitude, fact that was confirmed using computer based calculations and will be further explained in the following analysis. In the table below is shown the changes between the true and expected multitude of prim numbers in the first 12 silver intervals. The expected or active multitude of prime numbers is calculated using the relation cancelling will reasonably be born in the statistical calculations of (2.1) fewer prospective (active) erasers. This way, however, there will be, from (2.1), more prime numbers than the real ones, because x does not exist (i.e. virtual), hence it did not accomplish statistical erasing, like β did. So to reverse the result x needs to be added to become from virtual to real. The same will be true for the right boarder ζ of ( ) Similarly, because the right boarders e.g. ζ, η of ( ) are possibly, in general, not overlapping, there needs to be anew an increase in the active multitude s K , for the same reason as before, by one more unit. In total, by increasing There will be an increment for every band of this ideal case [of (4.7)] by So with 2 further erasers, for every band, the over-covering mentioned before is certain. Because in this way a new ideal distribution is born "that it is certain to realise all the missing erasing and even more". That leads to an inequality. A more analytical proof of the above is given here. Let δ , that is we achieve the realization of the statistically predicted coincidences from the relationship (4.7), and on the other hand simultaneously we succeed that these erasers to have greater density than the real density 1 j q . So finally in the virtual silver interval we will surely have more write-offs than the real ones. In others words we will have something that required from the asking inequality". This proof, combined with the definition of the virtual silver interval of precise calculations of prime numbers (due to the equal-distribution of the erasers of the bands as mentioned), will clarify the analysis below, of the inequalities mentioned and justified before, with a different additional way.
It is observed that in the relation (4.5) it was proved that 4 M s d q > . So now, the two additional erasers of relation (4.14) must be distributed in more than 4   ρ < < will be true, which led to the two relations (4.15) that will be used below, because it was shown that they are generic and true for every case. On the grounds that, even if the observation ρ < < will arise, which drive to the same conclusion that the above relation (4.15) is true. As a result the true probability will be defined respectively by (4.7) using the true density s ρ which satisfies the relation (4.15), meaning: And now obviously due to the (4.15) it is true that: However, in this way an interesting scenario occurs, a sequence of inequalities: ( )  ν ν + were inserted in brackets (…) exactly where they were missing, which creates a more enhanced inequality.
The last arose after the erasing of the equal numerators and denominators.
Consequently for the true function ( ) p ν , that defines the exact number of prime numbers in the random silver interval M δ , the result will be: ( )  however 2 g = was chosen because that was the one that allowed the erasing of sequential fractions, which at the end led to the proof of (4.18) that in turn proved to be sufficient for the calculation of the multitude of twin prime numbers, as it will be shown.
It was proven before, that an increment of ideal erasers (active) of every band for a mean multitude ( ) ( ) Based on this correction (4.16) forms the previous Table 1 as follows.
It is observed from Table 2 that there is indeed an important correction from an initial active multitude of 278 expected prime numbers, to 257.3 now. Meaning an error of approximately 1.4% from 6.5% that was before in Table 1.
On the subject of the twin pair, which was studied after Introduction, in Section 1, it was seen by relation (1.1) that a specific event with a probability p with regards to its appearance in a single repetition, in a multitude of N independent repetitions, will have a mean multitude of twin pairs, (which will be precise in infinity repetitions) equal to: The precise relation (4.19) derives from the independency of repetitions in a multitude of Ν − 1 boarders among these.
The known theorem of prime numbers that dictates a logarithmic distribution [1] [3] [6] is essentially a statistical theorem. To be exact, in this paper's Statistics, based on the relations (2.1), (4.18) that were proven and will be utilized below, it will additionally be validated that "the catholic (random) selection of a prime number a q (that was named in the beginning of this paper) in an also catholic (randomly) selected silver interval M δ , does not give the catholic information (that was also named in the beginning of this paper) that the probability of appearance of the next prime number b q is changing in the very same silver interval according to the distance of its position from a q ". The useful meaning of this catholic capacity is that the prime numbers are distributed in the catholic silver interval in such way that it does not statistically favor, after all in their infinity multitude, neither them getting closer nor away from each other. One of the initial reasons for this is that the creation of the prime numbers from the tracker does not produce any logical suggestion that will neither dictate them getting closer to each other in the same silver interval, which statistically would proposal of generic validity in N), then this catholic event would prevent the probative method of creation of (2.1) to ignore it, and thus it would be disclosed in that way. Consequently, since in mathematics nothing happens without a reason, the distribution of the prime numbers in a random (or otherwise catholic) selected silver interval will be the one that dictates the catholic relation (2.1).
The catholic relations (4.18), (4.18a), but also any of their improvements, do not change the above conclusion because they are objective inequalities. Meaning the reason is that this inequalities of catholic validity are exclusively due to the disorder of calculations that creates the border between two sequential catholic selected silver intervals, which however does not change the way the erasers that create the relation (2.1) work; using the equal-distribution of the probabilities of the prime numbers in this catholic selected silver interval, but this border prevents their precise calculations due to the non-overlapping of the limits of the erasers bands with the limits of the silver intervals, as it was already mentioned.

Proof
It is known, (and easy to be shown), that the minimum distance ij L between two multiples of the random (meaning; catholic selected) prime numbers i q , is no. The proof for this is based on the fact that the prime numbers have multiples independent per two and that means that the same will happen with the square of the prime numbers which define the boundaries of the silver intervals.
Thus these boundaries do not catholically affect on the distribution of prime numbers in the catholic silver interval. So during the selection of a random (representative catholic selected) silver interval it will occur inductively for its boundaries to also be independent events regarding the prime number multiples of the subsequence of this catholic selected interval. Thus these boundaries will be catholically independent regarding the position that the prime numbers will appear in random silver interval.
We can finally prove that not only the probability P ν is constant within a randomly selected silver interval M δ (or in others words P ν independent of the position of a random candidate prime number M a ν δ ∈ ) but and the varia- For the clarification of the definition inductively, that was used previously, it should be explained that the creation of prime numbers, by the procedure that was used to prove the relation (2.1) (which drives to creation of prime numbers in silver intervals and based on the independent multiples of prime numbers of their subsequence) starts without any overruling of the above proof form the beginning of the set of the silver intervals. Thus it is necessary at this point to investigate the inaugural structure at the proof above. Indeed, the first silver interval ( ) δ does not includes any element and so therefore the sentence "the numbers 2, 3 are not multiples of a non-existent prime number" is true. Afterwards, it is noticed that the multiples of these initial prime numbers 2 and 3 are independent from each other in the set N and thus the prime numbers that will be selected, based on the same selection rule, in the next silver intervals 1 δ , 2 δ will be always, according to the previous proving procedure, independent events of the probability theory and so on inductively.
Additionally, the boarders of the following silver interval that is created from the squares of the prime numbers 2, 3 do not contribute any logical catholic suggestion that will affect the divisibility both in this silver interval and in a catholic (randomly) selected one below. In this way we understand that the squares of prime numbers which define the silver intervals (because they define their boundaries) will be catholically independent events of the established probability theory.
Hence, according to the Proposition of catholic Information, that was mentioned in the Introduction of this article, there is no catholic Information that will force the infinity multitude of prime numbers to being thicken or dilute in a statistically random (CS) silver interval, and thus (based on the obvious axiomatic logical Proposition of catholic Information, in the introduction) one can execute (for infinity multitude of prime numbers) accurate statistical calculation (exactly because this multitude is infinite) using the established probability theory. And this is something that we will make below. In other words the total appearance of prime numbers in N is same with one of the infinite results-games (with infinite rotations for each game) of an ideal roulette that in every game has on its rotating disc (in every rotation) the multiples of those prime numbers which inductively produced using the previous prime numbers, which were found in the same way as before, and so on. Saying "random" way we mean "without additional catholic information except the one dictated by the relation (2.1)". The reason is that the action of inequalities (4.18) and (4.18a) etc are only preventing the accurate calculation of the multitude of the prime numbers in a silver interval and do not affect the catholic power of (2.1) for the reason stated in previous paragraphs.
Returning to the problem of the twin prime numbers, according to everything that was stated, the system of the aforementioned relations (2.1) and (4.18) will dictate a catholic distribution of prime numbers, where the frequency of their appearance will remain constant in the same silver interval. Thus the positions of prime numbers are per two independent from each other and is reduced in a rate dictated by the two relations (2.1) and (4.18) from the one silver interval to the next. Thus the frequency of appearance of prime numbers is changing only between silver intervals and not in the same silver interval, and all these are de- for the tallying of the twin prime numbers, that form the independent events of the appearance of prime numbers, meaning the independent prime numbers, gives: Hence, the infinity multitude of the silver intervals one has: However, the second part of (4.21) becomes infinity because the brackets are the result of the subtraction of a finite multitude of terms from the known harmonic series, that as proven in the introduction of unit 1, it becomes infinity: Therefore, it was proven that the wanted multitude of the twin prime numbers in the infinite multitude of natural numbers will also be infinite.
Without damaging the generality, assuming the twin pairs from the left, that is assuming the twin pairs of every new prime natural number candidate in the random silver interval M δ (with probability P ν ) with the one smaller (on its left) also new candidate prime (and hence its twin) natural number (with probability 1 P ν − ) it is found: Taking into consideration, in every M δ for Μ > 0, that the terms are of even multitude. The obvious reason for the clarification is that the disorder, that is caused by the relation above (in the tally R of twin numbers) also refers to the boarders of the sequential silver intervals, due to the fact that for these the probability P ν changes. So for the relations (4.21) there is: which is a condition able to make the multitude of twin prime pairs in the set N of natural numbers infinity and thus concludes this proof, since it is true that left R = ∞ . The question here is why it was not taken into consideration the causal datum that "the twin prime pairs of two successive natural numbers is excluded".
The answer is that this calculation ignores this kind of causal data, because it is based only on the possibility of twin pairs. If, however, on the base of a seemingly stricter proving procedure one considers this datum (information), then the above calculation is repeated using the set of the odd natural integers. In The reader can easily note that the steps from the relation (4.20) to the relation (4.22) can once more verify that the multitude of the prime numbers in the set of the natural N is infinite, if 2 P ν is replaced by P ν in (4.20) calculating now the multitude of prime natural numbers.
Furthermore, it will be shown that the probability of appearance of a prime number in a silver interval tends towards zero, meaning: From the inequality (4.18) that was shown here and the relation (2.9) that was shown in Section 2; the relation (4.23) is immediately proven: This relation is very important, because it states according to the definition given by Shannon that the information that the prime numbers enclose is infinite. Considering the set of natural numbers N quantified, in the sense that both its definition and all of its properties can be supported on the numbering, without breaking the quantum "1" that defines on the basis of the numbering all of the properties of set N, one concludes that the information will also be quantified in N. Indeed, since there is no prime number factorised, then according to the Shannon definition every new prime number in N defines the information ( )  which has the property to not be broken in a sum of two or more information terms smaller than the prime number. Consequently, in accordance with (4.24) every new prime number creates a new form of information, because it will not be analysed in a previous prime numbers information, additionally this information tends towards infinity. In the author's book "The twins of infinity and the Riemann hypothesis", Ziti ("Ζήτη") publications, it is defined as quantum q the least quantity of a quality Q. And as quality Q is defined a notifier set of properties, that are altered only during the break of quantum q. For example, the quantum of the quality "water" will be the "molecule of water". And therefore the q will be the physical unit of set Q with which one counts the repetitions of Q in a phenomenon in physics (or in mathematics). For instance, according to these the quantum of the Euclidean space-time of the Special Theory of Relativity will be four-dimensional and elementary hypercube of the space-time with an edge length equal to the length of Planck (10 −35 m). Furthermore, based on the above definition, every well-defined pure physical size, will owing to be quantified and that will be the definition of the pure physical size. And it is truly very charming the question, whether on the infinite information of prime numbers is paradoxically mirrored an infinite and unchanged hyper-verse of events.

The Dark Paths of Infinity and a Second Solution
This is at last the final step of this research. It will define one more kind of intervals on an axis of natural number, because their inconceivable length will be named dark or gloomy intervals. The first of these will be the interval of 25 natural numbers: is the last prime number, meaning the greatest prime number that will be included in the immediately previous gloomy interval, meaning it will be the one that will be included in the second which is ( ) It should be clarified here that the above events of divisibility X, Y have catholic probability (frequency of their appearance) equal to 1 j q , exactly as it is required to apply with absolute accuracy the probability theory below. The reason is that the "random" j q is an event (during its appearance) independent regarding the appearance of every prime factor of the "random"

( )
i Ω , hence it will also be an event independent from the natural number α, β, according always to the definitions of the independency of events in the probability theory that was mentioned in the beginning. It is about the catholic or random selection exactly as required by an accurate calculation in an infinity multitude of events.
The probabilities of the two prime events A and B will satisfy the relations (4.18) shown before in Section 4. That means that the following relations will be true: The events X, Y will have equal probabilities: ( ) ( ) The relations (5.6) have an obvious cause. For example if 3 j q = then the probability of a random natural number to be divided by 3 is 1/3, since one of the three natural numbers is third-multiple, meaning the integer multiple of the prime number 3. Similarly the probability to be divided as an integer a random natural number (as above the unspecified α, β) by j q is 1 j q , since one in the j q -multitude natural numbers is multiple of j q . Based on (5.5) one gets: ( ) ( )   to be destroyed by the appearance of j q , that from now on will be called the potential gloomy twin of ( ) Z i interval". Provided that 258 j q > , which is not of interest, because for this proof the infinite last gloomy intervals ( ) Z i are enough i.e. with 2 i > .
What will happen now if a second prime number j q λ ± is created? The relation (5.9) now with j q λ ± in the place of j q it will once more be true, with the main difference now being that except from the change in the probability of (5.9) there is another additional probability for the creation of a twin, because there will be the probability of the new of the new j q λ ± to become neighbors with the already existing prime number j q in the gloomy interval ( ) Z i and so on.
In conclusion there is a more powerful inequality for the relation of tallying the mean multitude of the twin pairs in ( )