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A Possible Solution for Hilbert's Unsolved 8th Problem: Twin Prime Conjecture

We measure whether there are numerous pairs of twin primes (hereafter referred to as twin prime pairs) according to the prime number inferred by sieve of Eratosthenes. In this study, we reveal at least three additional twin prime pairs increased as n is increased by 1, while setting (6n+5)**2 as the range for estimating twin prime pairs. As a result, we prove the Twin Prime Conjecture proposed by de Polignac in 1849. That is, there are numerous twin prime pairs, indicating that there are numerous prime number p for each natural number k by making p + 2k as prime number for the case of k = 1.


Introduction
We measure whether there are numerous pairs of twin primes (hereafter referred to as twin prime pairs) by employing the concept of inferring the prime number proposed by sieve of Eratosthenes. While we set (6n+5) 2 as the range for estimating twin prime pairs, we reveal at least three additional twin prime pairs as n (positive integer) is increased by 1. Thus, we confirm the Twin Prime Conjecture made by de Polignac in 1849 by proving that there are numerous prime number p for each natural number k by making p +2k as prime number as k = 1.
In 1990, Hilbert raised 23 mathematical problems at the 1900 International Mathematical Conference held in Paris [2]. The Twin Prime Puzzle (i.e. whether are numerous twin primes existed) is still listed in the 8 th question, even though Twin Prime Conjecture is proposed by de Polignac in 1849.
To my understanding, these are only three questions unsolved completely among these 23 mathematical problems [2]. As for the recent studies in terms of Hilbert's unsolved 8 th problems: Twin Prime Conjecture, Y. Zhang proves that bounded gaps between primes are all less than 70 million [7].
After establishing a bound of 70,000,000 on the narrowest gap between primes [7], there are tremendous interests in employing Zhang's argument to lower the bound. T. Tao and dozens of mathematicians worked together to cut the bound down from 70,000,000 to the final value of 246 via an online "Polymath" project as well as more traditional research channels [6]. However, these efforts are based on the conjecture about prime intervals [1]. In other words, the proposition seems changed from whether there are numerous twin prime pairs existed to another proposition: how much the upper limit of the distance between two prime numbers is [3], [5], [6].
Instead of narrowing down the distance between two prime numbers, we focus on the Twin Prime Conjecture: whether there are numerous twin prime pairs. In addition, the question of whether there exist numerous twin prime pairs has been one of the great open questions in number theory for more than a century.
In this study, we endeavor to prove whether there are numerous twin prime pairs. While setting (6n+5) 2 as the range 1 for estimating twin prime pairs in accordance 1 According to sieve of Eratosthenes [4], N would be prime number if N would not be divided by the prime number smaller than √ . Thus, while setting √ = 6n-1 (prime number) or 6n+1 (prime number), N would be a prime number close to (6n+5) 2 (i.e. (6(n+1)-1) 2 ), which would not be divided with Sieve of Eratosthenes [4], we prove that at least three additional twin prime pairs would exist as n (positive integer) is increased by 1. In other words, twin prime pairs would be numerous as n is increased from 1 to ∞ ∞ ∞ ∞. . . .

Associate twin prime (ATP) pairs defined and employed
In this study, we define the pairs (6n-1, 6n+1) as ATP (associate twin prime) pairs whose the first number and the second number in ATP pairs would not be included the multiple of 2 and 3 as shown in Table 1.
In addition, the ATP pairs (6n-1, 6n+1) would contain all twin prime pairs except the pair (3, 5) such as twin prime pair (5, 7) as n =1, twin prime pair ( In other words, if and only if ATPn a and ATPn b are all prime numbers, then the ATP pair (ATPn) would be regarded as a twin prime pair.
The above concern is to explore whether we are able to find the evidence of numerous twin prime pairs due to that the number of twin prime pair would be increased continuously as the selected incremental range expanded continuously. Thus, we would explain the selection for the incremental range in the following section.

Selection for Incremental Range
By calculating the number of ATP pairs within a certain range, we are able to explain why the number of ATP pairs would increase as the range is increased incrementally (i.e. the range is increased incrementally from (6n+5) 2 to (6(n+1)+5) 2 for n=1 to ∞ ∞ ∞ ∞). That is, we argue that the number of twin prime pairs would be increased infinitely when selecting the range is expanded infinitely. Then, the conjecture of the twin prime would be confirmed in this study.
As mentioned above, if and only if ATPn a and ATPn b are all prime numbers, then the ATP pair (ATPn) would be regarded as a twin prime pair. Thus, the Twin Prime pairs might be estimated as long as we could examine whether all these ATPn a and ATPn b less than (6n+5) 2 (i.e. (ATP(n+1) a ) 2 = (6(n+1)-1) 2 ) would not be divided by any prime number less than ATP(n+1) a such as 6n+1 less than 6n+5 (i.e. 6(n+1)-1). Thus, the following context is set (6n+5) 2 as the selection for the incremental range in this study, which would be convenient for us to explore whether there are unlimited twin prime pairs in this study.
Thus, if the twin prime numbers for ATPGn would be increased more than 1 as long as n is incremented by 1 each time, then we are able to prove Twin Prime Conjecture, i.e. (i.e. there are numerous twin primes pairs existed).
We then calculate the increasing rate of ATPGn (i.e. ATPGn%) as n is incremented by 1, and the formula is shown as below.
As mentioned above in section 2, the first number and the second number of an ATP pair defined in this study would be started from the integer 5 (i.e. the smallest first number of ATP pair (6n-1, 6n+1) would be 5). As a result, the Associate Twin Prime Group (ATPG0) would be equal to 3 for n=0 (i.e. the ATP pairs (5, 7), (11, 13), and (17, 19)) as the range smaller than (6n+5) 2 = 25 as n=0.
By employing the above equation, we would get the ATPG0=3 (i.e. the initial value = 47, and so on.
In addition, we would further measure the equation: TPEn = ATPGn‧TPRn, where TPEn is the estimated number of twin prime pairs accumulated to the range less than (6n+5) 2 for n = 1 to unlimited, ATPGn is the number of ATP pairs as integer number less than (6n+5) 2 for n=1 to unlimited, and TPRn is the probability of ATP pairs being twin prime pairs within the range (6n+5) 2 for n=1 to unlimited. As a result, we are able to measure the TPEn (The estimated number of Twin Prime pairs accumulated from the range (6n+5) 2 as n=0 (initial value =3) to the range (6n+5) 2 as n= unlimited.
In this study, we define TPEn as the estimated number of twin prime pairs accumulated to the range less than (6n+5) 2 for n = 1 to unlimited, ATPGn as the number of ATP pairs as integer number less than (6n+5) 2 for n=1 to unlimited, and TPRn as the probability of ATP pairs being twin prime pairs within the range (6n+5) 2 for n=1 to unlimited. Then, by measuring the equation: TPEn = ATPGn‧TPRn, we are able to measure the TPEn from the range (6n+5) 2 as n=1 2 to the range (6n+5) 2 as n= unlimited. In other words, we would measure whether the twin prime pairs would be increased unlimitedly as the incremental range, (6n+5) 2 increased from 1 to limited in this study. Thus, we would measure the probability of ATP pairs being twin prime pairs within the range (6n+5) 2 for n=1 to unlimited in the following section.

Figure 1. Excluding Twin Prim pairs from ATP pairs for Prime numbers 5 and 7
In Figure 1, the numbers in black boxes are excluded as prime numbers due to these numbers would be divided by 5 or 7. As either the first number or second number in an ATP pair (6n-1, 6n+1) is not prime numbers, this ATP pair would not be a Twin Prime pair. We also find that excluding twin prims pairs from ATP pairs would be rather regular, i.e. the probability of excluding twin prime pair from ATP pairs would be 2/x.
According to the screening approach proposed by sieve of Eratosthenes, some ATP pairs would be excluded from twin prime pairs, since the multiples of prime numbers smaller than √ are shown in the first numbers or second numbers in ATP pairs.
In fact, some of the first number and second number in ATP are not prime numbers instead of composite numbers, which would not be included as twin prime pair. For example, the number 35 in ATP pair would be excluded as well as the numbers 175 and 245 in ATP pairs would be excluded as well due to that these numbers are composite numbers.
In this study, the probability of Case I: ATPn a and ATPn b are all prime numbers would be #$8& 9: (The ratio of twin prime pairs over ATP pairs; Twin Prime Ratio) could be measured as shown below. We also find that excluding twin prims pairs from ATP pairs would be rather regular, i.e. the probability of excluding twin prime pair from ATP pairs would be 2/x.
In addition, x would be the first number or second number in ATP pair (ATPn a , ATPn b ).
However, we argue that there are four cases for ATP pair (ATPn a , ATPn b ) including Case I, (prime number, prime number), Case II, (prime number, composite number), Case III, (composite number, prime number), and Case IV (composite number, composite number). Thus, we are able to measure various probabilities for Cases II -IV as shown below. #$8& 9 = 1 − for Case II: ATPn a is prime number but ATPn b is not prime numbers. #$8& : = 1 − ; for Case III: ATPn a is not prime number but ATPn b is prime numbers. #$8& = 1 for Case IV: ATPn a and ATPn b are all not prime numbers.

Evidence for the existence of infinite Twin Prime Pairs
In summary, the conservative estimation of Twin Prime pairs could be measured as that the number of ATP Group (ATPGn) multiplied by the probability of Twin Prime pairs over ATP pairs (TRPn).
Thus, TPEn = ATPGn‧TPRn , where TPEn is the estimated number of twin prime pairs accumulated to the range less than (6n+5) 2 for n = 1 to unlimited, ATPGn is the number of ATP pairs as integer number less than (6n+5) 2 for n=1 to unlimited, and TPRn is the probability of ATP pairs being twin prime pairs within the range (6n+5) 2 for n=1 to unlimited. We then employ #$8& 9: instead of TPRn a , TPRn b , and TPRn 0 to measure the above equation, and the numbers of Twin Prime pairs estimated would be rather conservative as well. In order to prove Twin Prime Conjecture, the above math equation would be simplified as below. , we would get would be equal to 3n +3 as shown below.
After we differentiate the above equation, we would determine that this conservative estimator is an incremental function U U (3n+3)=3 The above inference indicates that three Twin Prime pairs would be increased as n is increased by 1.
Thus, we then prove the Twin Prime Conjecture proposed by de Polignac in 1849.
That is, there are numerous Twin Prime pairs, indicating that there are numerous prime number p for each natural number k by making p +2k as prime number as well for the case of k = 1.
In addition, we measure TPEn (i.e. TPEn = ATPGn‧TPRn, while setting TPRn = #$8& 9: ), TPAn (i.e. the actual twin prime pairs), and TPEn-S (i.e. the more conservative estimation for measuring TPE due to excluding non-negative term as well as setting TRRn = #$8& 9: ) for n=1 to 50 as shown in Table 2. In addition, we observe that TPEn is less than TPAn as well as TPEn-S is far less than TPAn. Even so, we confirm that there are numerous twin prime pairs, as the results shown for TPEn-S and TPEn. In addition, TPAn (Actual TP numbers) might have the following two concerns. One is the probabilities measured by #$8& 9 , #$8& : , #$8& , [\ #$8& 9: might be depended on various cases shown in (ATPn a , ATPn b ), and the other is to include the non-negative term as mentioned above.
In addition, we plot the trend for TPEn, TPAn, and TPEn-S. In fact, while setting (6n+5) 2 as the incremental range for measuring the numbers of twin prime pairs, we observe that increasing convex curves are shown for TPEn and TPAn, and increasing linear curve is shown for TPn-S due to that TPE-S=3n+3. Moreover, TPAn is much higher than TPEn-S and TPEn. While n is approaching unlimited, we are able to confirm numerous twin prim pars for TPEn-S. However, the numbers of twin prime pairs measured by TPEn-S are far less than those measured by TPEn. Likewise, the numbers of twin prime pairs measured by TPEn are far less than those measured by TPAn, indicating that confirming numerous twin prime pairs existed is rather persuasive in this study. (2) Excluding the non-negative term In fact, the above estimation would be regarded as a rather conservative estimation, and this conservative estimation mainly comes from two parts. One is that we employ #$8& 9: instead of TPRn a , TPRn b , and TPRn 0 do to measure TPRn-S. The other is that we simplify mathematical presentation by excluding the non-negative In addition, if we employ #$8& 9: , TPRn a , TPRn b , or TPRn 0 depended on the actual condition occurred as well as include a non-negative term in the mathematical equation, the numbers of Twin Prime pairs would be increased than the conservative TPEn-S estimation measured above as well as the result would be closer to the actual result. As a result, how to get the estimated result close to the actual result would be worthwhile for further studies.

Conclusion
We extend the sieve of Eratosthenes to calculate whether there are numerous numbers of Twin Prime pairs. In order to make our revealed result reliable even persuaded, we use very conservative estimation.
In this study, we prove that there are at least 3 additional twins prime pairs increased while setting (6n+5) 2 as the range of estimating the additional increase of twins prime pairs as n is increased by 1.
Furthermore, although estimating the twin prime pairs conservatively, we still prove that there are numerous twin prime pairs, since there are at least 3 additional twin prime pairs increased while setting (6n+5) 2 as the range of estimating the additional increase of twins prime pairs as n is increased by 1 for n =1, 2, ………..∞ ∞ ∞ ∞.
We then confirm that additional twin prime pairs would be increased as n is increased as well. Due to that n would be increased from 1 to ∞ ∞ ∞ ∞, twin prime pairs could be increased to infinite as well. That it, Hilbert's unsolved 8 th problems: Twin Prime Conjecture has been resolved in this study.
In fact, Twin Prime Conjecture solved might be based on two main concerns. One is to delete the non-twin prime pairs over ATP pairs would be regular (i.e. 2/x shown in context). The other is the incremental range from (6n+5) 2 to (6(n+1)+5) 2 for n =1, 2, 3,….infinite (i.e. the incremental range increased could be regarded as regular as well). We argue that Twin Prime Conjecture solved might be due to these two concerns are taken into account deliberately in the study.
Due to that, the estimation of whether there are infinite numbers of Twin Prim pairs is rather conservative and even simplified, the more accurate estimation would be taken into account for further studies.