Let x be positive integer and prime counting function π(x) denote the number of primes not greater than x. Then prime number theorem states that there is a limit such that

$\underset{x\to \infty }{\lim }\frac{\pi \left(x\right)}{\left(\frac{x}{\log x}\right)}=1$.(2.1)

This limit means the relative error between π(x) and x/logx approaches 0 as x grows without bound, that is, π(x) is asymptotically expressed as

$\pi \left(x\right)~\frac{x}{\log x}$.(2.2)

It is asymptotic distribution law of primes.

The prime number theorem can also be written as an approximation for π(x) as follows

$\pi \left(x\right)\approx Li\left(x\right)$,(2.3)

where

$Li\left(x\right)={\displaystyle \underset{2}{\overset{x}{\int }}\frac{\text{d}t}{\log t}}$(2.4)

is the logarithmic integral and it has an equivalent asymptotic series such that

$\begin{array}{c}Li\left(x\right)\approx \frac{x}{\log x}{\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\frac{k!}{{\left(\log x\right)}^k}}\\ =\frac{x}{\log x}\left(1+\frac{1}{\log x}+\frac{2}{{\left(\log x\right)}^2}+\cdots \right).\end{array}$(2.5)

Definition 2.1 Let P_n denote the n-th prime. Then P_i + P_k is called a Goldbach number generated by P_n if i ≤ n, k ≤ n. L_n is called the largest strong Goldbach number generated by P_n if L_n is an even number such that every even number from 4 to L_n is the sum of two primes not greater than P_n but L_n + 2 is not such a sum. Every even number from 4 to L_n is called a strong Goldbach number generated by P_n.

By Definition 2.1, we can find all Goldbach numbers generated by a given prime, for example, all Goldbach numbers generated by P₁₁ = 31 are 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 58, 60, 62. But 56 = 54 + 2 is not Goldbach number generated by P₁₁ = 31. The smallest Goldbach number is 4 and the largest Goldbach number is 62 but every even number from 4 to 54 is strong Goldbach number generated by P₁₁ = 31 and 54 is the largest strong Goldbach number generated by P₁₁ = 31 in the example. The first 50 largest strong Goldbach numbers are listed as follows

4, 6, 10, 14, 18, 26, 30, 38, 42, 42, 54, 62, 74, 74, 90, 90, 90, 108, 114, 114, 134, 134, 146, 162, 172, 180, 186, 186, 218, 222, 230, 240, 240, 254, 258, 270, 270, 290, 290, 290, 330, 348, 348, 366, 366, 366, 398, 398, 410, 410.

More largest strong Goldbach numbers generated by primes less than 10⁷ can be found in [12] . Figure 1 in [11] shows distributions of P_n and L_n for 1000000 ≤ n ≤ 100000000.

Numerical evidence for the first 50 largest strong Goldbach numbers presents that there is an observed fact that L_n ≤ L_n+1, and such distribution characteristic can be seen more clearly from figure 3 in [9] to show distribution of L_n for n ≤ 200. The figure makes us realize that distribution of L_n is a step-type curve with upward trend. We have the following theorem to show the rationality of general existence for such steps.

Theorem 2.2 L_n ≤ L_n+1 for every n > 0.

Proof. Let L_n be the largest strong Goldbach number generated by P_n. Then L_n must be a strong Goldbach number generated by P_n+1 by Definition 2.1. Therefore, it is impossible that L_n > L_n+1 for any n > 0. Suppose L_n = L_n+1 for all n > 0. Then if there is a counterevidence to this hypothesis then this hypothesis does not hold. Since there are many counterevidences to this hypothesis such as L₂₁₇₂₂ = 491586 < L₂₁₇₂₃ = 491612. Hence this hypothesis does not hold. Suppose L_n < L_n+1 for all n > 0. Then if there is a counterevidence to this hypothesis then this hypothesis does not hold. Since there are many counterevidences to this hypothesis such as L₂₁₇₂₆ = L₂₁₇₂₇ = 491612. Hence this hypothesis does not hold. Thus L_n ≤ L_n+1 for every n > 0 and the theorem holds.

Although there are infinitely many largest strong Goldbach numbers generated by infinitely many primes because L_n one-to-one corresponds to P_n, we can not confirm if there are infinitely many steps arising from L_n ≤ L_n+1. In order to consider whether there are infinitely many such steps, we have the following definitions.

Definition 2.3 Every step in distribution curve of L_n is called a Goldbach step.

Definition 2.4 For a given Goldbach step, W is called width of the Goldbach step if W = (n2 + 1) – n1, where n1 is n-value at the starting point of the Goldbach step and n2 is n-value at the finishing point of the Goldbach step but n2 + 1 is n-value at the starting point of next Goldbach step. H is called height of the Goldbach step if H = L_n1 for the Goldbach step, where n1 is n-value at the starting point of the Goldbach step.

By Definition 2.3 and Definition 2.4, W ≥ 1 for all n > 0 and every Goldbach step must be formed by one largest strong Goldbach number or more consecutive largest strong Goldbach numbers, thus, every L_n for n > 0 must belong to a Goldbach step. For example, L₁₂₈₆₉ = 275466 < L₁₂₈₇₀ = 275706 < L₁₂₈₇₁ = 276132 means there is a Goldbach step L₁₂₈₇₀ = 275706 whose height is L₁₂₈₇₀ = 275706 and width is (n2 +1) – n1 = (12870 + 1) – 12870 = 1. Another example is L₂₁₇₂₂ = 491586 < L₂₁₇₂₃ = L₂₁₇₂₄ = L₂₁₇₂₅ = L₂₁₇₂₆ = L₂₁₇₂₇ = 491612 < L₂₁₇₂₈ = 491624. It means there is a Goldbach step L₂₁₇₂₃ = L₂₁₇₂₄ = L₂₁₇₂₅ = L₂₁₇₂₆ = L₂₁₇₂₇ = 491612 whose height is L₂₁₇₂₃ = 491612 and width is (n2 + 1) – n1 = (21727 + 1) – 21723 = 5.

Remark 2.5 Theorem 2.2 implies general existence of Goldbach steps, that is, there is no a largest strong Goldbach number generated by prime not to belong to a Goldbach step. It means the prime sequence $\left\{P_n;P_n<P_n{}_{+1},n=1,2,3,\cdots \right\}$ to be an infinite sequence will generate the largest strong Goldbach number sequence $\left\{L_n;L_n\le L_n{}_{+1},n=1,2,3,\cdots \right\}$ to be also an infinite sequence and show general existence of Goldbach steps. But we have not known if there are infinitely many Goldbach steps based on existence of the largest strong Goldbach number sequence $\left\{L_n;L_n\le L_n{}_{+1},n=1,2,3,\dots \right\}$.

Lemma 2.6 If L_n approaches infinity as n grows without bound, then Goldbach conjecture is true.

Proof. By Definition 2.1, every even number from 4 to L_n for a given prime P_n is a strong Goldbach number generated by P_n, thus, all even numbers from 4 to L_n also must be Goldbach numbers generated by P_n. It means if L_n approaches infinity as n grows without bound then all even numbers greater than 2 will be Goldbach numbers and Goldbach conjecture is true.

Theorem 2.7 If Goldbach conjecture is true, then there are infinitely many Goldbach steps.

Proof. If there is a finite number of Goldbach steps, then there must exist a Goldbach step to be the last Goldbach step which has infinite width to prevent the occurrence of more Goldbach steps, and the Goldbach step with infinite width must also stop the increase of value of largest strong Goldbach number generated by prime so that there is no asymptotic result such that L_n approaches infinity as n grows without bound to imply Goldbach conjecture as Lemma 2.6 describes, thus, Goldbach conjecture is not true. Since this conclusion contradicts condition in this theorem. Hence if Goldbach conjecture is true then there is no the last Goldbach step whose width is infinite but there are infinitely many Goldbach steps and the theorem holds.

Let Q(n) be Goldbach step counting function and denote the number of Goldbach steps among the first n largest strong Goldbach numbers. Then we can get the observed value of Q(n) for 1 ≤ n ≤ 4000000000 in this paper. We had proposed an approximation Q′(n) for Q(n) as follows [11]

$Q^{\prime }\left(n\right)=Li\left(n\right)+\frac{n}{\log n\log \log n}$,(2.6)

where

$Li\left(n\right)={\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{\log t}}$,

and the asymptotic series for Li(n) is that

$Li\left(n\right)\approx \frac{n}{\log n}{\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\frac{k!}{{\left(\log n\right)}^k}}=\frac{n}{\log n}\left(1+\frac{1}{\log n}+\frac{2}{{\left(\log n\right)}^2}+\cdots \right)$.(2.7)

Taking the first three terms in (2.7) [13] , (2.6) becomes

$Q^{\prime }\left(n\right)\approx \frac{n}{\log n}\left(1+\frac{1}{\log n}+\frac{2}{{\left(\log n\right)}^2}+\frac{1}{\log \log n}\right)$.(2.8)

In this paper, (2.6) is improved as

$Q^{\prime }\left(n\right)=Li\left(n\right)+\frac{n}{\log n}\left(\frac{1}{\log \log n}+\frac{1}{4{\left(\log \log n\right)}^2}\right)$.(2.9)

Taking the first three terms in (2.7), above approximation becomes

$Q^{\prime }\left(n\right)\approx \frac{n}{\log n}\left(1+\frac{1}{\log n}+\frac{2}{{\left(\log n\right)}^2}+\frac{1}{\log \log n}+\frac{1}{4{\left(\log \log n\right)}^2}\right)$.(2.10)

We can give relative error between Q(n) and Q′(n) in Table 1 , where Q(n) is Goldbach step counting function, Q′(n) is the value predicted by (2.10) and π(n) is prime counting function.

Based on above numerical evidence, one can expect the relative error between Q(n) and Q′(n) will approach 0 as n grows without bound. It means that there is a limit as follows

$\underset{n\to \infty }{\lim }\frac{Q\left(n\right)}{Q^{\prime }\left(n\right)}=1$.(2.11)

By (2.9), the limit becomes

$\underset{n\to \infty }{\lim }\frac{Q\left(n\right)}{Li\left(n\right)+\frac{n}{\log n}\left(\frac{1}{\log \log n}+\frac{1}{4{\left(\log \log n\right)}^2}\right)}=1$,(2.12)

and we have the following theorem.

Theorem 2.8 If $\underset{n\to \infty }{\lim }\frac{Q\left(n\right)}{Li\left(n\right)+\frac{n}{\log n}\left(\frac{1}{\log \log n}+\frac{1}{4{\left(\log \log n\right)}^2}\right)}=1$, then $\underset{n\to \infty }{\lim }\frac{Q\left(n\right)}{\left(\frac{n}{\log n}\right)}=1$.

Proof. Using (2.7), $Li\left(n\right)+\frac{n}{\log n}\left(\frac{1}{\log \log n}+\frac{1}{4{\left(\log \log n\right)}^2}\right)$ can be written as

$\frac{n}{\log n}{\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\frac{k!}{{\left(\log n\right)}^k}}+\frac{n}{\log n}\left(\frac{1}{\log \log n}+\frac{1}{4{\left(\log \log n\right)}^2}\right)$.(2.13)

Considering asymptotic series in (2.13), we see that the k-th term approaches higher order infinity than the (k + 1)-th term as n grows without bound in the asymptotic series because there is the following limit.

$\underset{n\to \infty }{\lim }\frac{\frac{\left(k+1\right)!n}{{\left(\log n\right)}^{k+2}}}{\frac{k!n}{{\left(\log n\right)}^{k+1}}}=\underset{n\to \infty }{\lim }\frac{k+1}{\log n}=0$.(2.14)

Since the first term in the asymptotic series for Li(n) is n/logn, there are two limits for two additional terms as follows

$\underset{n\to \infty }{\lim }\frac{\frac{n}{\log n}\frac{1}{\log \log n}}{\frac{n}{\log n}}=\underset{n\to \infty }{\lim }\frac{1}{\log \log n}=0$,(2.15)

$\underset{n\to \infty }{\lim }\frac{\frac{n}{\log n}\frac{1}{4{\left(\log \log n\right)}^2}}{\frac{n}{\log n}}=\underset{n\to \infty }{\lim }\frac{1}{4{\left(\log \log n\right)}^2}=0$.(2.16)

Since it is assumed that

$\underset{n\to \infty }{\lim }\frac{Q\left(n\right)}{Li\left(n\right)+\frac{n}{\log n}\left(\frac{1}{\log \log n}+\frac{1}{4{\left(\log \log n\right)}^2}\right)}=1$,

by (2.13), (2.14), (2.15) and (2.16) we have

$\underset{n\to \infty }{\lim }\frac{Q\left(n\right)}{\left(\frac{n}{\log n}\right)}=1$.(2.17)

Hence the theorem holds.

Corollary 2.9 If $\underset{n\to \infty }{\lim }\frac{Q\left(n\right)}{Li\left(n\right)+\frac{n}{\log n}\left(\frac{1}{\log \log n}+\frac{1}{4{\left(\log \log n\right)}^2}\right)}=1$, then Goldbach conjecture is true.

Proof. Suppose $\underset{n\to \infty }{\lim }\frac{Q\left(n\right)}{Li\left(n\right)+\frac{n}{\log n}\left(\frac{1}{\log \log n}+\frac{1}{4{\left(\log \log n\right)}^2}\right)}=1$. Then, by Theorem 2.8 we have the following limit.

$\underset{n\to \infty }{\lim }\frac{Q\left(n\right)}{\left(\frac{n}{\log n}\right)}=1$.(2.18)

It means Q(n) is asymptotically expressed as

$Q\left(n\right)~\frac{n}{\log n}$.(2.19)

From (2.19) we see that Q(n) approaches infinity as n grows without bound if n/logn approaches infinity as n grows without bound. Since it is obvious that n/logn approaches infinity as n grows without bound. Hence Q(n) approaches infinity as n grows without bound. It means that there are infinitely many Goldbach steps. Since height H_i = L_n1-i of the i-th Goldbach step must be smaller than height H_i+1 = L_n1-(i+1) of the (i + 1)-th Goldbach step by Theorem 2.2 and definition 2.4. Hence L_n1-i < L_n1-(i+1) for all i > 0 so that L_n1-i approaches infinity as i grows without bound. Since the sequence $\left\{L_n{}_{1-i};L_n{}_{1-i}<L_n{}_{1-(i+1)},i=1,2,3,\cdots \right\}$ is a subsequence of the largest strong Goldbach number sequence {L_n; L_n ≤ L_n+1, n = 1, 2, 3, …}. Hence the result that L_n1-i approaches infinity as i grows without bound must lead to the result that L_n approaches infinity as n grows without bound. By Lemma 2.6, Goldbach conjecture is true and the corollary holds.

Limit (2.18) and asymptotic expression (2.19) mean that the relative error between Q(n) and n/logn approaches 0 as n grows without bound, thus, both the two results are same as statement of the prime number theorem as follows

$\underset{n\to \infty }{\lim }\frac{\pi \left(n\right)}{\left(\frac{n}{\log n}\right)}=1$,(2.20)

and

$\pi \left(n\right)~\frac{n}{\log n}$.(2.21)

Formulas (2.18), (2.19), (2.20) and (2.21) mean asymptotic distribution law of Goldbach steps is same as asymptotic distribution law of primes described by prime number theorem if Goldbach conjecture is true so that Goldbach steps are infinite by Theorem 2.7. Therefore, we say that Goldbach steps have like-prime nature and the nature will bring us a chance to study a kind of special natural numbers which seem to have such like-prime nature to represent the existence of Goldbach steps.

We have seen that it is clear that Goldbach steps have like-prime nature if there are infinitely many Goldbach steps because asymptotic distribution law of Goldbach steps is same as asymptotic distribution law of primes described by prime number theorem. Therefore, it is necessary that we should find a kind of special natural numbers for representing existence of all Goldbach steps and these numbers can well embody like-prime nature of Goldbach steps.

Definition 3.1 Let P_n denote the n-th prime. Then natural number n = n1 is called a like-prime if n = n1 is n-value at the starting point of a Goldbach step.

Remark 3.2 By Definition 3.1, the first like-prime is 1 and every like-prime greater than 1 satisfies L_n1–1 < L_n1 but the first like-prime 1 does not satisfy L_n1–1 < L_n1 because there is no L_n–1 = L₀. By Definition 3.1, we also see that if a natural number n is not like-prime then the natural number must satisfy L_n–1 = L_n, which means if L_n–1 = L_n then L_n must be on a Goldbach step whose width is greater than 1 but n > n1 for the step.

By Definition 3.1 we can find known like-primes. Let n1-i denote the i-th like-prime. Then the largest like-prime is known as n1-68459494 = 999999987 for n ≤ 10⁹ and the first 50 like-primes are listed as follows

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 18, 19, 21, 23, 24, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 38, 41, 42, 44, 47, 49, 51, 52, 54, 59, 61, 63, 64, 67, 69, 70, 71, 72, 77, 81, 83, 86.

More like-primes can be found in [12] . Q(n) in Table 1 gives the number of like-primes for n ≤ 10², 10³, 10⁴, 10⁵, 10⁶, 10⁷, 10⁸, 10⁹.

Lemma 3.3 n1-i < n1-(i + 1) for all i > 0.

Proof. Since n1-i is n-value at the starting point of the i-th Goldbach step and n1-(i + 1) is n-value at the starting point of the (i + 1)-th Goldbach step by Definition 2.4. Hence n1-i < n1-(i + 1) for all i > 0 by Theorem 2.2 and the lemma holds.

Remark 3.4 Definition 3.1 and Lemma 3.3 mean that the existence of all Goldbach steps is represented by the like-prime sequence {n1-i; n1-i < n1-(i + 1), i = 1, 2, 3, …} but we do not know if the sequence is an infinite sequence.

Theorem 3.5 If Goldbach conjecture is true, then the like-prime sequence {n1-i; n1-i < n1-(i + 1), i = 1, 2, 3, …} is an infinite sequence.

Proof. By Theorem 2.7, If Goldbach conjecture is true then there are infinitely many Goldbach steps. Since every Goldbach step is represented by a like-prime n1 and n1-i < n1-(i + 1) for all i > 0 by Lemma 3.3. Hence if Goldbach conjecture is true then there are infinitely many like-primes n1 such that n1-i < n1-(i + 1) for all i > 0, thus, the like-prime sequence {n1-i; n1-i < n1-(i + 1), i = 1, 2, 3, …} is an infinite sequence and the theorem holds.

Note there is a converse theorem of Theorem 3.5.

Theorem 3.6 If the like-prime sequence {n1-i; n1-i < n1-(i + 1), i = 1, 2, 3,…} is an infinite sequence, then Goldbach conjecture is true.

Proof. Suppose the like-prime sequence {n1-i; n1-i < n1-(i + 1), i = 1, 2, 3, …} is an infinite sequence. Then there are infinitely many Goldbach steps because every n1 denotes a Goldbach step by Definition 3.1. Since L_n1-i < L_n1-(i+1) for all i > 0 by Definition 2.4. Hence L_n1-i approaches infinity as i grows without bound to lead to the result that L_n approaches infinity as n grows without bound. By Lemma 2.6 Goldbach conjecture is true and the theorem holds.

Remark 3.7 Theorem 3.6 means if it can be proven that the like-prime sequence {n1-i; n1-i < n1-(i + 1), i = 1, 2, 3, …} is an infinite sequence, then Goldbach conjecture is true.

Definition 3.8 Largest strong Goldbach number L_n is called a largest strong Goldbach number with distinct value if n = n1.

Remark 3.9 Definition 3.8 means all L_n1, which are heights of Goldbach steps, are a kind of special largest strong Goldbach numbers to satisfy L_n1-i < L_n1-(i+1) for all i > 0, where L_n1-i is the i-th largest strong Goldbach number with distinct value. Thus, there is the sequence of largest strong Goldbach numbers with distinct values {L_n1-i; L_n1-i < L_n1-(i+1), i = 1, 2, 3, …}. By Definition 3.8, approximate and asymptotic distribution laws of Goldbach steps (2.9) and (2.19) are also approximate and asymptotic distribution laws of largest strong Goldbach numbers with distinct values.

By Definition 3.8 we can find known largest strong Goldbach numbers with distinct values. The largest L_n1-i is L_n1-68459494 = L_999999987 = 45603524304 for n ≤ 10⁹ and the first 50 largest strong Goldbach numbers with distinct values are listed as follows

4, 6, 10, 14, 18, 26, 30, 38, 42, 54, 62, 74, 90, 108, 114, 134, 146, 162, 172, 180, 186, 218, 222, 230, 240, 254, 258, 270, 290, 330, 348, 366, 398, 410, 434, 440, 474, 522, 528, 566, 570, 614, 630, 634, 650, 680, 686, 722, 794, 822.

More largest strong Goldbach numbers with distinct values can be found in [12] and Q(n) in Table 1 gives the number of largest strong Goldbach numbers with distinct values for n ≤ 10², 10³, 10⁴, 10⁵, 10⁶, 10⁷, 10⁸, 10⁹.

Lemma 3.10 L_n1-i < L_n1-(i+1) for all i > 0.

Proof. Since L_n1-i is height of the i-th Goldbach step and L_n1-(i+1) is height of the (i + 1)-th Goldbach step by Definition 2.4. Hence L_n1-i < L_n1-(i+1) for all i > 0 and the lemma holds.

Remark 3.11 Definition 3.8 and Lemma 3.10 mean that the existence of all Goldbach steps is represented by the sequence of largest strong Goldbach numbers with distinct values {L_n1-i; L_n1-i < L_n1-(i+1), i = 1, 2, 3, …} but we do not know if the sequence is an infinite sequence.

Theorem 3.12 If Goldbach conjecture is true, then the sequence of largest strong Goldbach numbers with distinct values {L_n1-i; L_n1-i < L_n1-(i+1), i = 1, 2, 3, …} is an infinite sequence.

Proof. By Theorem 2.7, If Goldbach conjecture is true then there are infinitely many Goldbach steps. Since every Goldbach step is represented by a largest strong Goldbach number with distinct value and L_n1-i < L_n1-(i+1) for all i > 0 by Lemma 3.10. Hence if Goldbach conjecture is true then there are infinitely many largest strong Goldbach numbers with distinct values such that L_n1-i < L_n1-(i+1) for all i > 0, that is, the sequence of largest strong Goldbach numbers with distinct values {L_n1-i; L_n1-i < L_n1-(i+1), i = 1, 2, 3, …} is an infinite sequence and the theorem holds.

Note there is a converse theorem of Theorem 3.12.

Theorem 3.13 If the sequence of largest strong Goldbach numbers with distinct values {L_n1-i; L_n1-i < L_n1-(i+1), i = 1, 2, 3, …} is an infinite sequence, then Goldbach conjecture is true.

Proof. Suppose the sequence of largest strong Goldbach numbers with distinct values {L_n1-i; L_n1-i < L_n1-(i+1), i = 1, 2, 3, …} is an infinite sequence. Then there are infinitely many Goldbach steps because L_n1 denotes height of Goldbach step by Definition 2.4 and Definition 3.8. Since L_n1-i < L_n1-(i+1) for all i > 0 by Lemma 3.10. Hence L_n1-i approaches infinity as i grows without bound to lead to the result that L_n approaches infinity as n grows without bound. By Lemma 2.6 Goldbach conjecture is true and the theorem holds.

Remark 3.14 Theorem 3.13 means if it can be proven that the sequence of largest strong Goldbach numbers with distinct values {L_n1-i; L_n1-i < L_n1-(i+1), i = 1, 2, 3, …} is an infinite sequence, then Goldbach conjecture is true. The sequence of largest strong Goldbach numbers with distinct values seems to be similar to the like-prime sequence.

Because n1-i < n1-(i + 1) for all i > 0 by Lemma 3.3, there must exist gap between like-primes as gap between primes does. Thus, we have the following definition.

Definition 3.15 g_i is called the i-th like-prime gap if g_i = n1-(i + 1) – n1-i.

By Definition 3.15, the first 100 like-prime gaps are listed as follows

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 3, 2, 2, 1, 2, 5, 2, 2, 1, 3, 2, 1, 1, 1, 5, 4, 2, 3, 2, 3, 2, 2, 1, 5, 4, 1, 9, 4, 2, 4, 5, 2, 1, 2, 3, 2, 3, 3, 5, 1, 1, 2, 6, 1, 8, 1, 6, 1, 9, 2, 5, 10, 2, 3, 6, 1, 3, 3, 2, 4, 3, 5, 3, 3, 1, 1, 1, 1, 8.

There are some natural numbers to become length of gap between like-primes such as 1, 2, 3, 4, 5, 6, 8, 9, 10 among the first 100 like-prime gaps. More like-prime gaps can be found in [12] . By Definition 3.1 we see that if there is a natural number n greater than 1 such that L_n–1 = L_n then the natural number n is not a like-prime, thus, there are k – 1 consecutive natural numbers not to be like-prime between n1-i and n1-(i + 1) if g_i = k because like-prime gap is defined as g_i = n1-(i + 1) – n1-i. Considering k = 1, there is no natural number between n1-i and n1-(i + 1). Thus, we conjecture the number of consecutive natural numbers between n1-i and n1-(i + 1) can be arbitrarily large, that is, length of gap between like-primes can be arbitrarily large. If Goldbach conjecture is true, by Theorem 3.5 there are infinitely many like-primes to satisfy n1-i < n1-(i + 1). Therefore, by Definition 3.15 there are infinitely many like-prime gaps and we have the following result.

$n=1+{\displaystyle \underset{i=1}{\overset{\infty }{\sum }}{g}_i}$.

Let N be a natural number greater than 1. Then there is a sequence including N – 1 consecutive composite numbers N! + 2, N! + 3, …, N! + N because N! + 2 = 2(N!/2 + 1), N! + 3 = 3(N!/3 + 1), …, N! + N = N(N!/N + 1) all are composite numbers. Therefore, gap between primes can be arbitrarily large so that every positive even number can become length of a prime gap. Let p denote prime. Then Polignac’s conjecture states that there are infinitely many primes p such that p + 2k is also prime for every natural number k [14] , and specially, there are infinitely many primes p such that p + 2k is also prime for k = 1, which is twin prime conjecture. There is a strong form of twin prime conjecture, that is, an asymptotic distribution law of twin primes akin to the prime number theorem is a special case of the first Hardy-Littlewood conjecture [15] .

Let π₂(x) denote the number of primes p ≤ x such that p + 2 is also prime. Define twin prime constant C₂ as [16]

$C_2={\displaystyle \underset{p\ge 3}{\prod }\left(1-\frac{1}{{\left(p-1\right)}^2}\right)}={\displaystyle \underset{p\ge 3}{\prod }\frac{p\left(p-2\right)}{{\left(p-1\right)}^2}}\approx 0.660161815\cdots$.(3.1)

Then there is a special case of the first Hardy-Littlewood conjecture such that

${\pi }_2\left(x\right)\approx 2C_2{\displaystyle \underset{2}{\overset{x}{\int }}\frac{\text{d}t}{{\left(\log t\right)}^2}}$,

${\pi }_2\left(x\right)~2C_2\frac{x}{{\left(\log x\right)}^2}$,(3.2)

in the sense that the quotient of the two expressions approaches 1 as x grows without bound [17] . Although (3.2) to be a strong form of twin prime conjecture has not been proven, the conjecture seems certain to be true. Obviously, if (3.2) holds then twin prime conjecture is true. Table 2 gives the number of twin primes and the values predicted by the Hardy-Littlewood formula [15] .

Definition 3.16 A like-prime n1 is called a twin like-prime if n1 + 1 is also like-prime.

By Definition 3.16, we can find known pairs of twin like-primes and the first 50 pairs of twin like-primes are listed as follows

(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 8), (8, 9), (11, 12), (12, 13), (18, 19), (23, 24), (24, 25), (25, 26), (26, 27), (29, 30), (30, 31), (31, 32), (34, 35), (35, 36), (41, 42), (51, 52), (63, 64), (69, 70), (70, 71), (71, 72), (95, 96), (105, 106), (129, 130), (148, 149), (149, 150), (159, 160), (168, 169), (175, 176), (213, 214), (240, 241), (241, 242), (242, 243), (243, 244), (267, 268), (268, 269), (269, 270), (280, 281), (321, 322), (322, 323), (333, 334), (334, 335), (345, 346), (368, 369), (369, 370).

More twin like-primes can be found in [12] and Q₂(n) in Table 3 gives the number of twin like-primes for n ≤ 10², 10³, 10⁴, 10⁵, 10⁶, 10⁷, 10⁸, 10⁹.

Conjecture 3.17 There are infinitely many like-primes n1 such that n1 + 1 is also like-prime.

Corollary 3.18 If Conjecture 3.17 is true, then Goldbach conjecture is true.

Proof. Since every pair of twin like-primes (n1, n1 + 1) corresponds to a Goldbach step with width to be 1, the infinitude of twin like-prime pairs means that there are infinitely many Goldbach steps with width to be 1 to lead to the existence of infinitely many Goldbach steps. By L_n1-(i+1) > L_n1-t for all i > 0, L_n1-t approaches infinity as i grows without bound to lead to the result that L_n approaches infinity as n grows without bound. The result implies Goldbach conjecture by Lemma 2.6 and the corollary holds.

Remark 3.19 Conjecture 3.17 can be called twin like-prime conjecture to correspond to twin prime conjecture. As there is a strong form of twin prime conjecture, which is a special case of the first Hardy-Littlewood conjecture, there is also a strong form of twin like-prime conjecture as the following discussion does.

Let Q₂(n) denote the number of twin like-primes among the first n positive integers to number the first n primes for generating largest strong Goldbach numbers and be also called twin like-prime counting function. Then we propose that there is an approximation ${Q^{\prime }}_2\left(n\right)$ for Q₂(n) such that

$Q_2\left(n\right)\approx {Q^{\prime }}_2\left(n\right)$,

${Q^{\prime }}_2\left(n\right)=2C_2{\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{{\left(\log t\right)}^2}}+2C_2\frac{n}{{\left(\log n\right)}^2}\left(\frac{1}{\log \log n}+\frac{1}{{\left(\log \log n\right)}^2}\right)$,(3.3)

where C₂ is twin prime constant (3.1).

Since ${\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{{\left(\log t\right)}^2}}=Li\left(n\right)+\frac{2}{\log 2}-\frac{n}{\log n}$ [15] , (3.3) becomes

$\begin{array}{c}{Q^{\prime }}_2\left(n\right)=2C_2\left(Li\left(n\right)-\frac{n}{\log n}\right)+2C_2\frac{2}{\log 2}\\ +2C_2\frac{n}{{\left(\log n\right)}^2}\left(\frac{1}{\log \log n}+\frac{1}{{\left(\log \log n\right)}^2}\right).\end{array}$(3.4)

There is the asymptotic series for Li(n) such that

$\begin{array}{c}Li\left(n\right)\approx \frac{n}{\log n}{\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\frac{k!}{{\left(\log n\right)}^k}}\\ =\frac{n}{\log n}\left(1+\frac{1}{\log n}+\frac{2}{{\left(\log n\right)}^2}+\frac{6}{{\left(\log n\right)}^3}+\cdots \right).\end{array}$(3.5)

Taking the first four terms in (3.5), by (3.4) we get

$\begin{array}{c}{Q^{\prime }}_2\left(n\right)\approx 2C_2\frac{n}{{\left(\log n\right)}^2}\left(1+\frac{2}{\log n}+\frac{6}{{\left(\log n\right)}^2}+\frac{1}{\log \log n}+\frac{1}{{\left(\log \log n\right)}^2}\right)\\ +2C_2\frac{2}{\log 2}.\end{array}$(3.6)

We can give relative error between Q₂(n) and ${Q^{\prime }}_2\left(n\right)$ as Table 3 shows. In the table, Q₂(n) is the number of twin like-primes, ${Q^{\prime }}_2\left(n\right)$ is the values predicted by formula (3.6) and π₂(n) is the number of twin primes.

Theorem 3.20 If $\underset{n\to \infty }{\lim }\frac{Q_2\left(n\right)}{2C_2{\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{{\left(\log t\right)}^2}+2C_2\frac{n}{{\left(\log n\right)}^2}\left(\frac{1}{\log \log n}+\frac{1}{{\left(\log \log n\right)}^2}\right)}}=1$, then

$\underset{n\to \infty }{\lim }\frac{Q_2\left(n\right)}{2C_2\frac{n}{{\left(\log n\right)}^2}}=1$.

Proof. As we know, there are two results such that

${\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{{\left(\log t\right)}^2}}=Li\left(n\right)+\frac{2}{\log 2}-\frac{n}{\log n}$,(3.7)

$\begin{array}{c}Li\left(n\right)\approx \frac{n}{\log n}{\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\frac{k!}{{\left(\log n\right)}^k}}\\ =\frac{n}{\log n}\left(1+\frac{1}{\log n}+\frac{2}{{\left(\log n\right)}^2}+\frac{6}{{\left(\log n\right)}^3}+\cdots \right).\end{array}$(3.8)

Let ${Q^{\prime }}_2\left(n\right)=2C_2{\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{{\left(\log t\right)}^2}}+2C_2\frac{n}{{\left(\log n\right)}^2}\left(\frac{1}{\log \log n}+\frac{1}{{\left(\log \log n\right)}^2}\right)$. By (3.7) and (3.8) we have

$\begin{array}{c}{Q^{\prime }}_2\left(n\right)\approx 2C_2\frac{n}{\log n}{\displaystyle \underset{k=1}{\overset{\infty }{\sum }}\frac{k!}{{\left(\log n\right)}^k}+2C_2\frac{2}{\log 2}}\\ +2C_2\frac{n}{{\left(\log n\right)}^2}\left(\frac{1}{\log \log n}+\frac{1}{{\left(\log \log n\right)}^2}\right).\end{array}$(3.9)

Considering asymptotic series in (3.9), we see that the k-th term approaches higher order infinity than the (k + 1)-th term as n grows without bound because of the existence of the following limit.

$\underset{n\to \infty }{\lim }\frac{2C_2\frac{\left(k+1\right)!n}{{\left(\log n\right)}^{k+2}}}{2C_2\frac{k!n}{{\left(\log n\right)}^{k+1}}}=\underset{n\to \infty }{\lim }\frac{k+1}{\log n}=0$.(3.10)

Since the first term of asymptotic series in (3.9) is

$2C_2\frac{n}{{\left(\log n\right)}^2}$,

there is a limit for constant term in (3.9) as follows

$\underset{n\to \infty }{\lim }\frac{2C_2\frac{2}{\log 2}}{2C_2\frac{n}{{\left(\log n\right)}^2}}=\frac{2}{\log 2}\underset{n\to \infty }{\lim }\frac{{\left(\log n\right)}^2}{n}=0$.(3.11)

For two additional terms in (3.9), we have the following limits.

$\underset{n\to \infty }{\lim }\frac{2C_2\frac{n}{{\left(\log n\right)}^2\left(\log \log n\right)}}{2C_2\frac{n}{{\left(\log n\right)}^2}}=\underset{n\to \infty }{\lim }\frac{1}{\log \log n}=0$,(3.12)

$\underset{n\to \infty }{\lim }\frac{2C_2\frac{n}{{\left(\log n\right)}^2{\left(\log \log n\right)}^2}}{2C_2\frac{n}{{\left(\log n\right)}^2}}=\underset{n\to \infty }{\lim }\frac{1}{{\left(\log \log n\right)}^2}=0$.(3.13)

Since it is assumed that

$\underset{n\to \infty }{\lim }\frac{Q_2\left(n\right)}{2C_2{\displaystyle {\int }_2^n\frac{\text{d}t}{{\left(\log t\right)}^2}+2C_2\frac{n}{{\left(\log n\right)}^2}\left(\frac{1}{\log \log n}+\frac{1}{{\left(\log \log n\right)}^2}\right)}}=1$,

by (3.7), (3.8), (3.9), (3.10), (3.11), (3.12) and (3.13) we get

$\underset{n\to \infty }{\lim }\frac{Q_2\left(n\right)}{2C_2\frac{n}{{\left(\log n\right)}^2}}=1$.

Hence the theorem holds.

Corollary 3.21 If $\underset{n\to \infty }{\lim }\frac{Q_2\left(n\right)}{2C_2{\displaystyle {\int }_2^n\frac{\text{d}t}{{\left(\log t\right)}^2}+2C_2\frac{n}{{\left(\log n\right)}^2}\left(\frac{1}{\log \log n}+\frac{1}{{\left(\log \log n\right)}^2}\right)}}=1$, then Goldbach conjecture is true.

Proof. Suppose $\underset{n\to \infty }{\lim }\frac{Q_2\left(n\right)}{2C_2{\displaystyle {\int }_2^n\frac{\text{d}t}{{\left(\log t\right)}^2}+2C_2\frac{n}{{\left(\log n\right)}^2}\left(\frac{1}{\log \log n}+\frac{1}{{\left(\log \log n\right)}^2}\right)}}=1$.

Then we have the following limit by Theorem 3.20.

$\underset{n\to \infty }{\lim }\frac{Q_2\left(n\right)}{2C_2\frac{n}{{\left(\log n\right)}^2}}=1$.(3.14)

It means there is an asymptotic expression as follows

$Q_2\left(n\right)~2C_2\frac{n}{{\left(\log n\right)}^2}$.(3.15)

Formula (3.14) means the relative error between Q₂(n) and 2C₂n/(logn)² approaches 0 as n grows without bound, and formula (3.15) means that Q₂(n) is asymptotically equal to 2C₂n/(logn)². It is obvious that result (3.15) means that there are infinitely many pairs of twin like-primes since 2C₂n/(logn)² approaches infinity as n grows without bound. By Conjecture 3.17 and Corollary 3.18, Goldbach conjecture is true and the corollary holds.

There are two forms of prime 3-tuplet such that (p, p + 2, p + 6) and (p, p + 4, p + 6), and also called prime triplet. It is conjectured that there are infinitely many primes p such that p + 2 and p + 6 are also primes. It is also conjectured that there are infinitely many primes p such that p + 4 and p + 6 are also primes. The conjectures are called triplet prime conjecture. There is a strong form of triplet prime conjecture to be a special case of the first Hardy-Littlewood conjecture as the following discussion does.

Let π₃(x) denote the number of primes p ≤ x such that p + 2 and p + 6 or p + 4 and p + 6 are also primes. Define a constant C₃ as [15]

$C_3=\frac{9}{2}{\displaystyle \underset{p\ge 5}{\prod }\frac{p^2\left(p-3\right)}{{\left(p-1\right)}^3}}\approx 2.858248596\cdots$.(3.16)

Then there is a special case of the first Hardy-Littlewood conjecture such that

${\pi }_3\left(x\right)\approx C_3{\displaystyle \underset{2}{\overset{x}{\int }}\frac{\text{d}t}{{\left(\log t\right)}^3}}$,

${\pi }_3\left(x\right)~C_3\frac{x}{{\left(\log x\right)}^3}$,(3.17)

in the sense that the quotient of the two expressions approaches 1 as x grows without bound. Obviously, if (3.17) holds then triplet prime conjecture is true and twin prime conjecture is also true because every prime triplet must include a pair of twin primes. There are two tables, Table 4 and Table 5 , to give the number of prime triplets and the values predicted by the Hardy-Littlewood formula [15] .

Because there are two forms, (p, p + 2, p + 6) and (p, p + 4, p + 6), to represent existence of prime triplets. Correspondingly, we have the following two definitions.

Definition 3.22 A like-prime n1 is called a triplet like-prime if n1 + 1 and n1 + 3 are also like-primes but n1 + 2 is not a like-prime.

By Definition 3.22, we can find known like-prime triplets of this form and the first 50 like-prime triplets of this form are listed as follows

(8, 9, 11), (12, 13, 15), (18, 19, 21), (26, 27, 29), (31, 32, 34), (35, 36, 38), (41, 42, 44), (51. 52, 54), (129, 130, 132), (269, 270, 272), (394, 395, 397), (397, 398, 400), (437, 438, 440), (472, 473, 475), (543, 544, 546), (666, 667, 669), (1192, 1193, 1195), (1381, 1382, 1384), (1434, 1435, 1437), (1874, 1875, 1877), (2036, 2037, 2039), (2612, 2613, 2625), (2847, 2848, 2850), (2998 2999, 3001), (3489, 3490, 3492), (3492, 3493, 3495), (3561, 3562, 3564), (3803, 3804, 3806), (4071, 4072, 4074), (4150, 4151, 4153), (4311, 4312, 4314), (4405, 4406, 4408), (4619, 4620, 4622), (4808, 4809, 4811), (4831, 4832, 4834), (5269, 5270, 5272), (5297, 5298, 5300), (5394, 5395, 5397), (5459, 5460, 5462), (5510, 5511, 5513), (5630, 5631, 5633), (5647, 5648, 5650), (5761, 5762, 5764), (5876, 5877, 5879), (5879, 5880, 5882), (5958, 5959, 5961), (6014, 6015, 6017), (6388, 6389, 6391), (6983, 6984, 6986), (7345, 7346, 7348).

More like-prime triplets of this form can be found in [12] and Q₃(n) in Table 6 gives the number of like-prime triplets of the form (n1, n1 + 1, n1 + 3) with n1 + 2 not to be like-prime for n ≤ 10², 10³, 10⁴, 10⁵, 10⁶, 10⁷, 10⁸, 10⁹.

Definition 3.23 A like-prime n1 is called a triplet like-prime if n1 + 2 and n1 + 3 are also like-primes but n1 + 1 is not a like-prime.

By Definition 3.23, we can find known like-prime triplets of this form and the first 50 like-prime triplets of this form are listed as follows

(9, 11, 12), (21, 23, 24), (27, 29, 30), (32, 34, 35), (49, 51, 52), (61, 63, 64), (67, 69, 70), (93, 95, 96), (127, 129, 130), (319, 321, 322), (331, 333, 334), (395, 397, 398), (400, 402, 403), (537, 539, 340), (662, 664, 665), (667, 669, 670), (1163, 1165, 1166), (1172, 1174, 1175), (1190, 1192, 1193), (1379, 1381, 1382), (1384, 1386, 1387), (1432, 1434, 1435), (1646, 1648, 1649), (1880, 1882, 1883), (2037, 2039, 2040), (2101, 2103, 2104), (2179, 2181, 2182), (2532, 2534, 2535), (2570, 2572, 2573), (2617, 2619, 2620), (3184, 3186, 3187), (3487, 3489, 3490), (3490, 3492, 3493), (3631, 2633, 3634), (3643, 3645, 3646), (4031, 4033, 4034), (4069, 4071, 4072), (4147, 4149, 4150), (4406, 4408, 4409), (4448, 4450, 4451), (4832, 4834, 4835), (5270, 5272, 5273), (5325, 5327, 5328), (5392, 5394, 5395), (5416, 5418, 5419), (5642, 5644, 5645), (5677, 5679, 5680), (5759, 5761, 5762), (5877, 5879, 5880), (5959, 5961, 5962).

More like-prime triplets of this form can be found in [12] and Q₃(n) in Table 7 gives the number of like-prime triplets of the form (n1, n1 + 2, n1 + 3) with n1 + 1 not to be like-prime for n ≤ 10², 10³, 10⁴, 10⁵, 10⁶, 10⁷, 10⁸, 10⁹.

Conjecture 3.24 There are infinitely many triplet like-primes n1 defined by Definition 3.22.

Corollary 3.25 If Conjecture 3.24 is true, then Goldbach conjecture is true.

Proof. Since every like-prime triplet (n1, n1 + 1, n1 + 3) with n1 + 2 not to be like-prime must include a pair of twin like-primes to correspond to existence of a Goldbach step with width to be 1, the infinitude of such like-prime triplets means that there are infinitely many Goldbach steps with width to be 1 to lead to the existence of infinitely many Goldbach steps. By L_n1-(i+1) > L_n1-t for all i > 0, L_n1-t approaches infinity as i grows without bound to lead to a result that L_n approaches infinity as n grows without bound. The result implies Goldbach conjecture by Lemma 2.6 and the corollary holds.

Conjecture 3.26 There are infinitely many triplet like-primes n1 defined by Definition 3.23.

Corollary 3.27 If Conjecture 3.26 is true, then Goldbach conjecture is true.

Proof. Since every like-prime triplet (n1, n1 + 2, n1 + 3) with n1 + 1 not to be like-prime must include a pair of twin like-primes to correspond to existence of a Goldbach step with width to be 1, the infinitude of such like-prime triplets means that there are infinitely many Goldbach steps with width to be 1 to lead to the existence of infinitely many Goldbach steps. By L_n1-(i+1) > L_n1-t for all i > 0, L_n1-t approaches infinity as i grows without bound to lead to a result that L_n approaches infinity as n grows without bound. The result implies Goldbach conjecture by Lemma 2.6 and the corollary holds.

Remark 3.28 Triplet like-primes defined by Definition 3.22 can be called the first kind of triplet like-primes and triplet like-primes defined by Definition 3.23 can be called the second kind of triplet like-primes. Both Conjecture 3.24 and Conjecture 3.26 can be called triplet like-prime conjecture to correspond to triplet prime conjecture. As there is a strong form of triplet prime conjecture to be a special case of the first Hardy-Littlewood conjecture, there is also a strong form of triplet like-prime conjecture.

Let Q₃(n) denote the number of like-prime triplets among the first n positive integers to number the first n primes for generating largest strong Goldbach numbers. Then we propose that there is an approximation ${Q^{\prime }}_3\left(n\right)$ for Q₃(n) such that

$Q_3\left(n\right)\approx {Q^{\prime }}_3\left(n\right)$,

${Q^{\prime }}_3\left(n\right)=C_3{\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{{\left(\log t\right)}^3}}-C_3\frac{n}{{\left(\log n\right)}^3}\left(\frac{1}{3\log \log n}+\frac{1}{3{\left(\log \log n\right)}^2}\right)$,(3.18)

where C₃ is defined as a constant as follows

$C_3=\frac{9}{2}{\displaystyle \underset{p\ge 5}{\prod }\frac{p^2\left(p-3\right)}{{\left(p-1\right)}^3}}\approx 2.858248596\cdots$.(3.19)

This constant is just constant (3.16).

By ${\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{{\left(\log t\right)}^3}}=\frac{1}{2}Li\left(n\right)-\frac{n}{2{\left(\log n\right)}^2}-\frac{n}{2\log n}+\frac{1}{\log 2}+\frac{1}{{\left(\log 2\right)}^2}$ [15] , (3.18) becomes

$\begin{array}{c}{Q^{\prime }}_3\left(n\right)=C_3\left(\frac{1}{2}Li\left(n\right)-\frac{n}{2{\left(\log n\right)}^2}-\frac{n}{2\log n}\right)+C_3\left(\frac{1}{\log 2}+\frac{1}{{\left(\log 2\right)}^2}\right)\\ -C_3\frac{n}{{\left(\log n\right)}^3}\left(\frac{1}{3\log \log n}+\frac{1}{3{\left(\log \log n\right)}^2}\right).\end{array}$(3.20)

There is the asymptotic series for Li(n)/2 such that

$\begin{array}{c}\frac{1}{2}Li\left(n\right)\approx \frac{n}{2\log n}{\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\frac{k!}{{\left(\log n\right)}^k}}\\ =\frac{n}{2\log n}\left(1+\frac{1}{\log n}+\frac{2}{{\left(\log n\right)}^2}+\frac{6}{{\left(\log n\right)}^3}+\frac{24}{{\left(\log n\right)}^4}+\cdots \right).\end{array}$(3.21)

Taking the first five terms in the asymptotic series, (3.20) becomes

$\begin{array}{c}{Q^{\prime }}_3\left(n\right)\approx C_3\frac{n}{{\left(\log n\right)}^3}\left(1+\frac{3}{\log n}+\frac{12}{{\left(\log n\right)}^2}-\frac{1}{3\log \log n}-\frac{1}{3{\left(\log \log n\right)}^2}\right)\\ +C_3\left(\frac{1}{\log 2}+\frac{1}{{\left(\log 2\right)}^2}\right).\end{array}$(3.22)

We can give the relative error between Q₃(n) and ${Q^{\prime }}_3\left(n\right)$ as Table 6 shows. In the table, Q₃(n) is the number of the first kind of like-prime triplets and ${Q^{\prime }}_3\left(n\right)$ is the values predicted by formula (3.22) and π₃(n) is the number of prime triplets of the form (p, p + 2, p + 6).

We can give the relative error between Q₃(n) and ${Q^{\prime }}_3\left(n\right)$ as Table 7 shows. In the table, Q₃(n) is the number of the second kind of like-prime triplets, ${Q^{\prime }}_3\left(n\right)$ is the values predicted by formula (3.22) and π₃(n) is the number of prime triplets of the form (p, p + 4, p + 6).

From Table 6 and Table 7 we see both counted numbers of the first kind of like-prime triplets and the second kind of like-prime triplets seem to be close to the value predicted by formula (3.22), which means that it is reasonable to study Goldbach problem by introducing like-prime and like-prime gap.

Theorem 3.29 If $\underset{n\to \infty }{\lim }\frac{Q_3\left(n\right)}{C_3{\displaystyle {\int }_2^n\frac{\text{d}t}{{\left(\log t\right)}^3}-C_3\frac{n}{{\left(\log n\right)}^3}\left(\frac{1}{3\log \log n}+\frac{1}{3{\left(\log \log n\right)}^2}\right)}}=1$, then $\underset{n\to \infty }{\lim }\frac{Q_3\left(n\right)}{C_3\frac{n}{{\left(\log n\right)}^3}}=1$.

Proof. As we know, there are two results such that

${\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{{\left(\log t\right)}^3}=\frac{1}{2}Li\left(n\right)-\frac{n}{2{\left(\log n\right)}^2}-\frac{n}{2\log n}+\frac{1}{\log 2}+\frac{1}{{\left(\log 2\right)}^2}}$,(3.23)

$\begin{array}{c}Li\left(n\right)\approx \frac{n}{\log n}{\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\frac{k!}{{\left(\log n\right)}^k}}\\ =\frac{n}{\log n}\left(1+\frac{1}{\log n}+\frac{2}{{\left(\log n\right)}^2}+\frac{6}{{\left(\log n\right)}^3}+\cdots \right).\end{array}$(3.24)

Let ${Q^{\prime }}_3\left(n\right)=C_3{\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{{\left(\log t\right)}^3}}-C_3\frac{n}{{\left(\log n\right)}^3}\left(\frac{1}{3\log \log n}+\frac{1}{3{\left(\log \log n\right)}^2}\right)$. By (3.23) and (3.24) we have an expression for ${Q^{\prime }}_3\left(n\right)$ as follows

$\begin{array}{c}{Q^{\prime }}_3\left(n\right)\approx \frac{C_3}{2}\frac{n}{\log n}{\displaystyle \underset{k=2}{\overset{\infty }{\sum }}\frac{k!}{{\left(\log n\right)}^k}}+C_3\left(\frac{1}{\log 2}+\frac{1}{{\left(\log 2\right)}^2}\right)\\ -C_3\frac{n}{{\left(\log n\right)}^3}\left(\frac{1}{3\log \log n}+\frac{1}{3{\left(\log \log n\right)}^2}\right).\end{array}$(3.25)

Considering above asymptotic series, we see the k-th term approaches higher order infinity than the (k + 1)-th term as n grows without bound because there is a limit such that

$\underset{n\to \infty }{\lim }\frac{\frac{C_3}{2}\frac{\left(k+1\right)!n}{{\left(\log n\right)}^{k+2}}}{\frac{C_3}{2}\frac{k!n}{{\left(\log n\right)}^{k+1}}}=\underset{n\to \infty }{\lim }\frac{k+1}{\log n}=0$.(3.26)

Since the first term in above asymptotic series is

$C_3\frac{n}{{\left(\log n\right)}^3}$,

there is a limit for constant term as follows

$\underset{n\to \infty }{\lim }\frac{C_3\left(\frac{1}{\log 2}+\frac{1}{{\left(\log 2\right)}^2}\right)}{C_3\frac{n}{{\left(\log n\right)}^3}}=\left(\frac{1}{\log 2}+\frac{1}{{\left(\log 2\right)}^2}\right)\underset{n\to \infty }{\lim }\frac{{\left(\log n\right)}^3}{n}=0$.(3.27)

For above two additional terms, there are the following limits.

$\underset{n\to \infty }{\lim }\frac{C_3\frac{n}{3{\left(\log n\right)}^3\left(\log \log n\right)}}{C_3\frac{n}{{\left(\log n\right)}^3}}=\underset{n\to \infty }{\lim }\frac{1}{3\log \log n}=0$,(3.28)

$\underset{n\to \infty }{\lim }\frac{C_3\frac{n}{3{\left(\log n\right)}^3{\left(\log \log n\right)}^2}}{C_3\frac{n}{{\left(\log n\right)}^3}}=\underset{n\to \infty }{\lim }\frac{1}{3{\left(\log \log n\right)}^2}=0$.(3.29)

Since it is assumed that

$\underset{n\to \infty }{\lim }\frac{Q_3\left(n\right)}{C_3{\displaystyle {\int }_2^n\frac{\text{d}t}{{\left(\log t\right)}^3}-C_3\frac{n}{{\left(\log n\right)}^3}\left(\frac{1}{3\log \log n}+\frac{1}{3{\left(\log \log n\right)}^2}\right)}}=1$.(3.30)

By (3.23), (3.24), (3.25), (3.26), (3.27), (3.28),(3.29) and (3.30), we get

$\underset{n\to \infty }{\lim }\frac{Q_3\left(n\right)}{C_3\frac{n}{{\left(\log n\right)}^3}}=1$.

Hence the theorem holds.

Corollary 3.30 If $\underset{n\to \infty }{\lim }\frac{Q_3\left(n\right)}{C_3{\displaystyle {\int }_2^n\frac{\text{d}t}{{\left(\log t\right)}^3}-C_3\frac{n}{{\left(\log n\right)}^3}\left(\frac{1}{3\log \log n}+\frac{1}{3{\left(\log \log n\right)}^2}\right)}}=1$, then Goldbach conjecture is true.

Proof. Suppose $\underset{n\to \infty }{\lim }\frac{Q_3\left(n\right)}{C_3{\displaystyle {\int }_2^n\frac{\text{d}t}{{\left(\log t\right)}^3}-C_3\frac{n}{{\left(\log n\right)}^3}\left(\frac{1}{3\log \log n}+\frac{1}{3{\left(\log \log n\right)}^2}\right)}}=1$. Then we have the following limit by Theorem 3.29.

$\underset{n\to \infty }{\lim }\frac{Q_3\left(n\right)}{C_3\frac{n}{{\left(\log n\right)}^3}}=1$.(3.31)

It means there is an asymptotic expression as follows

$Q_3\left(n\right)~C_3\frac{n}{{\left(\log n\right)}^3}$.(3.32)

Formula (3.31) means the relative error between Q₃(n) and C₃n/(logn)³ approaches 0 as n grows without bound, and formula (3.32) means that Q₃(n) is asymptotically equal to C₃n/(logn)³. It is obvious that C₃n/(logn)³ approaches infinity as n grows without bound to lead to a result such that there are infinitely many triplet like-primes. Since every like-prime triplet must include a pair of twin like-primes. Hence there are infinitely many pairs of twin like-primes. By Conjecture 3.17 and Corollary 3.18, Goldbach conjecture is true and the corollary holds.

There is a form of prime 4-tuplet such that (p, p + 2, p + 6, p + 8), and also called a prime quadruplet. It can be conjectured that there are infinitely many primes p such that p + 2, p + 6, p + 8 are also primes. There is a strong form of the conjecture to be a special case of the first Hardy-Littlewood conjecture as the following discussion does.

Let π₄(x) denote the number of primes p ≤ x such that p + 2, p + 6 and p + 8 are also primes. Define a constant C₄ as [15]

$C_4=\frac{27}{2}{\displaystyle \underset{p\ge 5}{\prod }\frac{p^3\left(p-4\right)}{{\left(p-1\right)}^4}}\approx 4.151180864\cdots$.(3.33)

Then there is a special case of the first Hardy-Littlewood conjecture such that

${\pi }_4\left(x\right)\approx C_4{\displaystyle \underset{2}{\overset{x}{\int }}\frac{\text{d}t}{{\left(\log t\right)}^4}}$,

${\pi }_4\left(x\right)~C_4\frac{x}{{\left(\log x\right)}^4}$,(3.34)

in the sense that the quotient of the two expressions approaches 1 as x grows without bound. Obviously, if (3.34) holds then quadruplet prime conjecture is true and twin prime conjecture is also true because every prime quadruplet must include two pairs of twin primes. Table 8 gives the number of prime quadruplets of the form (p, p + 2, p + 6, p + 8) and the values predicted by the Hardy-Littlewood formula [15] .

Definition 3.31 A like-prime n1 is called a quadruplet like-prime if n1 + 1, n1 + 3 and n1 + 4 are also like-primes but n1 + 2 is not like-prime.

By Definition 3.31, we can find known like-prime quadruplets and the first 50 like-prime quadruplets are listed as follows

(8, 9, 11, 12), (26, 27, 29, 30), (31, 32, 34, 35), (394, 395, 397, 398), (666, 667, 669, 670), (2036, 2037, 2039, 2040), (3489, 3490, 3492, 3493), (4405, 4406, 4408, 4409), (4831, 4832, 4834, 4835), (5269, 5270, 5272, 5273), (5876, 5877, 5879, 5880), (5958, 5959, 5961, 5962), (8091, 8092, 8094, 8095), (10497, 10498, 10500, 10501), (10733, 10734, 10736, 10737), (11131, 11132, 11134, 11135), (12997, 12998, 13000, 13001), (13127, 13128, 13130, 13131), (14213, 14214, 14216, 14217), (17904, 17905, 17907, 17908), (27625, 27626, 27628, 27629), (30245, 30246, 30248, 30249), (31632, 31633, 31635, 31636), (39646, 39647, 39649, 30650), (41557, 41558, 41560, 41561), (41994, 41995, 41997, 41998), (43936, 43937, 43939, 43940), (44993, 44994, 44996, 44997), (46776, 46777, 46779, 46780), (46878, 46879, 46881, 46882), (46881, 46882, 46884, 46885), (64189, 64190, 64192, 64193), (66571, 66572, 66574, 66575), (69408, 69409, 69411, 69412), (74943, 74944, 74946, 74947), (75532, 75533, 75535, 75536), (83170, 83171, 83173, 83174), (91873, 91874, 91876, 91877), (111024, 111025, 111027, 111028), (115356, 115357, 115359, 115360), (120590, 120591, 120593, 120594), (123186, 123187, 123189, 123190), (127922, 127923, 127925, 127926), (129456, 129457, 129459, 129460), (130140, 130141, 130143, 130144), (137394, 137395, 137397, 137398), (138142, 138143, 138145, 138146), (139466, 139467, 139469, 139470), (141529, 141530, 141532, 141533), (143319, 143320, 143322, 143323).

More like-prime quadruplets can be found in [12] and Q₄(n) in Table 9 gives the number of like-prime quadruplets of the form (n1, n1 + 1, n1 + 3, n1 + 4) with n1 + 2 not to be like-prime for n ≤ 10², 10³, 10⁴, 10⁵, 10⁶, 10⁷, 10⁸, 10⁹.

Conjecture 3.32 There are infinitely many like-primes n1 such that n1 + 1, n1 + 3 and n1 + 4 are also like-primes but n1 + 2 is not like-prime.

Corollary 3.33 If Conjecture 3.32 is true, then Goldbach conjecture is true.

Proof. Since every like-prime quadruplet (n1, n1 + 1, n1 + 3, n1 + 4) with n1 + 2 not to be like-prime must include two pairs of twin like-primes to correspond to two Goldbach steps with width to be 1, the infinitude of like-prime quadruplets means that there are infinitely many pairs of twin like-primes. By Conjecture 3.17 and Corollary 3.18, Goldbach conjecture is true and the corollary holds.

Remark 3.34 Conjecture 3.32 can be called quadruplet like-prime conjecture to correspond to quadruplet prime conjecture. As there is a strong form of quadruplet prime conjecture, which is a special case of the first Hardy-Littlewood conjecture, there is also a strong form of quadruplet like-prime conjecture as the following discussion does.

Let Q₄(n) denote the number of quadruplet like-primes among the first n positive integers to number the first n primes for generating largest strong Goldbach numbers. Then we propose that there is an approximation ${Q^{\prime }}_4\left(n\right)$ for Q₄(n) such that

$Q_4\left(n\right)\approx {Q^{\prime }}_4\left(n\right)$,

${Q^{\prime }}_4\left(n\right)=C_4{\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{{\left(\log t\right)}^4}}-C_4\frac{n}{{\left(\log n\right)}^4}\left(\frac{1}{4\log \log n}+\frac{1}{4{\left(\log \log n\right)}^2}\right)$,(3.35)

where C₄ is defined as a constant as follows

$C_4=\frac{27}{2}{\displaystyle \underset{p\ge 5}{\prod }\frac{p^3\left(p-4\right)}{{\left(p-1\right)}^4}}\approx 4.151180864\cdots$.(3.36)

This constant is just constant (3.33). By

${\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{{\left(\log t\right)}^4}}=\frac{1}{6}Li\left(n\right)-\frac{n}{3{\left(\log n\right)}^3}-\frac{n}{6{\left(\log n\right)}^2}-\frac{n}{6\log n}+\frac{2}{3{\left(\log 2\right)}^3}+\frac{1}{3{\left(\log 2\right)}^2}+\frac{1}{3\log 2}$

[15] , (3.35) becomes

$\begin{array}{c}{Q^{\prime }}_4\left(n\right)=C_4\left(\frac{1}{6}Li\left(n\right)-\frac{n}{3{\left(\log n\right)}^3}-\frac{n}{6{\left(\log n\right)}^2}-\frac{n}{6\log n}\right)\\ +C_4\left(\frac{2}{3{\left(\log 2\right)}^3}+\frac{1}{3{\left(\log 2\right)}^2}+\frac{1}{3\log 2}\right)\\ -C_4\frac{n}{{\left(\log n\right)}^4}\left(\frac{1}{4\log \log n}+\frac{1}{4{\left(\log \log n\right)}^2}\right).\end{array}$(3.37)

There is the asymptotic series for Li(n)/6 such that

$\begin{array}{c}\frac{1}{6}Li\left(n\right)\approx \frac{n}{6\log n}{\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\frac{k!}{{\left(\log n\right)}^k}}\\ =\frac{n}{6\log n}\left(1+\frac{1}{\log n}+\frac{2}{{\left(\log n\right)}^2}+\frac{6}{{\left(\log n\right)}^3}+\frac{24}{{\left(\log n\right)}^4}+\frac{120}{{\left(\log n\right)}^5}+\cdots \right).\end{array}$ (3.38)

Taking the first six terms in the asymptotic series, (3.37) becomes

$\begin{array}{c}{Q^{\prime }}_4\left(n\right)\approx C_4\frac{n}{{\left(\log n\right)}^4}\left(1+\frac{4}{\log n}+\frac{20}{{\left(\log n\right)}^2}-\frac{1}{4\log \log n}-\frac{1}{4{\left(\log \log n\right)}^2}\right)\\ +C_4\left(\frac{2}{3{\left(\log 2\right)}^3}+\frac{1}{3{\left(\log 2\right)}^2}+\frac{1}{3\log 2}\right).\end{array}$(3.39)

We can give the relative error between Q₄(n) and ${Q^{\prime }}_4\left(n\right)$ as Table 9 shows. In the table, Q₄(n) is the number of like-prime quadruplets, ${Q^{\prime }}_4\left(n\right)$ is the values predicted by formula (3.39) and π₄(n) is the number of prime quadruplets.

Theorem 3.35 If $\underset{n\to \infty }{\lim }\frac{Q_4\left(n\right)}{C_4{\displaystyle {\int }_2^n\frac{\text{d}t}{{\left(\log t\right)}^4}-C_4\frac{n}{{\left(\log n\right)}^4}\left(\frac{1}{4\log \log n}+\frac{1}{4{\left(\log \log n\right)}^2}\right)}}=1$, then

$\underset{n\to \infty }{\lim }\frac{Q_4\left(n\right)}{C_4\frac{n}{{\left(\log n\right)}^4}}=1$.

Proof. As we know, there are two results such that

${\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{{\left(\log t\right)}^4}}=\frac{1}{6}Li\left(n\right)-\frac{n}{3{\left(\log n\right)}^3}-\frac{n}{6{\left(\log n\right)}^2}-\frac{n}{6\log n}+\frac{2}{3{\left(\log 2\right)}^3}+\frac{1}{3{\left(\log 2\right)}^2}+\frac{1}{3\log 2}$,

$\begin{array}{c}Li\left(n\right)\approx \frac{n}{\log n}{\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\frac{k!}{{\left(\log n\right)}^k}}\\ =\frac{n}{\log n}\left(1+\frac{1}{\log n}+\frac{2}{{\left(\log n\right)}^2}+\frac{6}{{\left(\log n\right)}^3}+\cdots \right).\end{array}$(3.40)

Let ${Q^{\prime }}_4\left(n\right)=C_4{\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{{\left(\log t\right)}^4}}-C_4\frac{n}{{\left(\log n\right)}^4}\left(\frac{1}{4\log \log n}+\frac{1}{4{\left(\log \log n\right)}^2}\right)$. By (3.40) we have

$\begin{array}{c}{Q^{\prime }}_4\left(n\right)\approx \frac{C_4}{6}\frac{n}{\log n}{\displaystyle \underset{k=3}{\overset{\infty }{\sum }}\frac{k!}{{\left(\log n\right)}^k}}+C_4\left(\frac{2}{3{\left(\log 2\right)}^3}+\frac{1}{3{\left(\log 2\right)}^2}+\frac{1}{3\log 2}\right)\\ -C_4\frac{n}{{\left(\log n\right)}^4}\left(\frac{1}{4\log \log n}+\frac{1}{4{\left(\log \log n\right)}^2}\right).\end{array}$(3.41)

Considering the asymptotic series in (3.41), we see the k-th term approaches higher order infinity than the (k + 1)-th term as n grows without bound. Therefore, we have the following limit.

$\underset{n\to \infty }{\lim }\frac{\frac{C_4}{6}\frac{\left(k+1\right)!n}{{\left(\log n\right)}^{k+2}}}{\frac{C_4}{6}\frac{k!n}{{\left(\log n\right)}^{k+1}}}=\underset{n\to \infty }{\lim }\frac{k+1}{\log n}=0$.(3.42)