Pingyuan Zhou
Beiyuan 35-210, Chengdu University of Technology, Chengdu, China.
DOI: 10.4236/apm.2026.168029   PDF    HTML   XML   6 Downloads   49 Views   Citations

Abstract

Bertrand theorem states that there is at least one prime in (x, 2x) for x > 1. Let P_n denote the n-th prime and take x = P_n. Then the theorem states that there is at least one prime in (P_n, 2P_n). It shows P_n+1 − P_n < P_n, which means that gap between primes is linearly controlled and supports infinitude of primes. Generalize the theorem into the prime index sequence {n}. Then the Bertrand-type theorem states there is at least one number n + k in (n, 2n) for n > 1 such that n + k is prime p and P_p is called double prime. It is obvious that the Bertrand-type theorem implies infinitude of double primes. An even number L_n is defined as the largest strong Goldbach number generated by P_n if 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. Since L_n ≤ L_n+1 for all n, L_n is a non-decreasing function and there exist growth points of L_n. If L_n−1 < L_n then n is a growth point of L_n and corresponding prime P_n is called a nontrivial prime to structure a growth of L_n. It is clear that the infinitude of nontrivial primes implies Goldbach conjecture. Comparing counted number of nontrivial primes with counted number of double primes, we can conjecture that there is at least one number n + k in (n, 2n) for n > 1 such that P_n+k is a nontrivial prime. The Bertrand-type conjecture has been verified up to n = 300,000. If it is proven then Goldbach conjecture is true. Comparing counted number of twin primes with counted number of double primes, we can conjecture that there is at least one number n + k in (n, 2n) for n > 1 such that P_n+k is a twin prime. The Bertrand-type conjecture has been verified up to n = 300,000. If it is proven then twin prime conjecture is true.

Keywords

Bertrand Theorem, Bertrand-Type Theorem for Double Prime, The Largest Strong Goldbach Number, Nontrivial Prime, Bertrand-Type Conjecture for Nontrivial Prime, Bertrand-Type Conjecture for Twin Prime, Root of Prime, Gap between Prime Roots, Goldbach Conjecture, Twin Prime Conjecture

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1. Introduction

The Goldbach conjecture states that every even number greater than 2 is the sum of two primes and the conjecture remains unsolved to this day. Of studies on the conjecture, main research results arise from circle method, sieve method and exceptional set method [1]-[6]. The twin prime conjecture states that there are infinitely many primes p such that p + 2 is also prime. The conjecture is a special case of Polignac conjecture, which states that there are infinitely many primes p such that p + 2k is also prime for every natural number k [7]. Hardy-Littlewood conjectured distribution of twin primes is asymptotically expressed as 2C₂x/(logx)², which is a special case of the first Hardy-Littlewood conjecture[1]. Although the conjecture has not been proven, it seems certain to be true. It presents a strong form for proving twin prime conjecture. An important research advance on the twin prime conjecture is that it was proven that there are infinitely many prime gaps with length bounded by N = 246 in 2013 [8]. This is a weak result of Polignac conjecture. It is obvious that Goldbach conjecture and twin prime conjecture are not the same type of problem. In order to transform Goldbach conjecture into a problem about infinitude of a kind of special primes, we tightened concept of traditional Goldbach number, that is, an even number is called a Goldbach number generated by a prime if the even number is the sum of two primes not greater than this prime. The covering boundary of such Goldbach numbers to consecutive even numbers is defined as the largest strong Goldbach number generated by a prime [9]-[13]. Let P_n denote the n-th prime and L_n denote the largest strong Goldbach number generated by P_n. Then L_n+1 ≥ L_n for all n. If L_n−1 < L_n then n is a growth point of L_n and corresponding P_n is definite as a nontrivial prime. It is clear that the infinitude of nontrivial primes implies Goldbach conjecture. Thus the Goldbach conjecture has been transformed into a problem about the infinitude of a kind of special primes as the twin prime conjecture does. What mathematical form can provide a common frame for studying the two conjectures? When we consider existence problems of nontrivial primes and twin primes in prime index interval, a Bertrand-type theorem for existence of double primes in prime index interval seems to be able to construct the frame, where P_n is called a double prime if n is prime p. Bertrand theorem states there is at least one prime in (x, 2x) for x > 1. Taking x = P_n, the theorem states there is at least one prime in (P_n, 2P_n). Generalize Bertrand theorem into the prime index sequence {n}. Then the Bertrand-type theorem states there is at least one number n + k in (n, 2n) for n > 1 such that n + k is prime p. If n is defined as root of prime P_n then the Bertrand-type theorem is presented more clearly as follows. Let n denote root of P_n and 2n denote root of P_2n. Then there is at least one prime root n + k in (n, 2n) for n > 1 such that n + k is double prime root, equivalently, there is at least one prime P_n+k in (P_n, P_2n) for n > 1 such that P_n+k is a double prime. The Bertrand-type theorem has been verified up to n = 300,000. Write the i-th double prime root as A(i). Then we have A(i + 1) − A(i) < A(i). It means that gap between double prime roots remains linearly controlled, which supports infinitude of double prime roots to show infinitude of double primes. If Bertrand theorem is established to show existence of primes in natural number interval, then Bertrand-type theorem is established to show existence of double primes in prime index interval. The Bertrand-type theorem for double prime is a standard system for solving existence problem of second order primes on n-axis. Comparing counted number of nontrivial primes on n-axis with counted number of double primes on n-axis, we have the following conjecture. Let n denote root of P_n and 2n denote root of P_2n. Then there is at least one prime root n + k in (n, 2n) for n > 1 such that n + k is a nontrivial prime root, equivalently, there is at least one prime P_n+k in (P_n, P_2n) for n > 1 such that P_n+k is a nontrivial prime. Write the i-th nontrivial prime root as B(i). Then we have B(i + 1) − B(i) < B(i), which means that gap between nontrivial prime roots remains linearly controlled. It supports infinitude of nontrivial prime roots to show infinitude of nontrivial primes and Goldbach conjecture is true. The Bertrand-type conjecture has been verified up to n = 300,000. Comparing counted number of twin primes on n-axis with counted number of double primes on n-axis, we have the following conjecture. Let n denote root of P_n and 2n denote root of P_2n. Then there is at least one prime root n + k in (n, 2n) for n > 1 such that n + k is a twin prime root, equivalently, there is at least one prime P_n+k in (P_n, P_2n) for n > 1 such that P_n+k is a twin prime. Write the i-th twin prime root as C(i). Then we have C(i + 1) − C(i) < C(i), which means gap between twin prime roots remains linearly controlled. It supports infinitude of twin prime roots to show infinitude of twin primes and twin prime conjecture is true. The Bertrand-type conjecture has been verified up to n = 300,000. If above two Bertrand-type conjectures are proven, then the Goldbach conjecture and the twin prime conjecture are true.

2. Bertrand-Type Theorem for Double Prime

2.1. Bertrand Theorem

Before prime number theorem was proven in 1896 [14] [15], Bertrand had made a conjecture while studying permutation groups in 1845, which is known as Bertrand’s postulate. The conjecture states that there is at least one prime in (x, 2x) for x > 1 [16]. Bertrand verified his conjecture up to x = 3,000,000. Chebyshev proved the conjecture using estimates of prime counting function and introducing Chebyshev functions in 1850 [17]. Thus, Bertrand theorem sometimes is called Bertrand-Chebyshev theorem. The first elementary proof of Bertrand’s postulate was given by Ramanujan using the Gamma function and leading to the notion of Ramanujan primes in 1919 [18]. Later, the proof was improved by 19-year-old Erdös based on combinatorial arguments in 1932 [19]. Euclid proved there are infinitely many primes by definition of prime but did not address how far apart consecutive primes can be. Take x = P_n. Then Bertrand’s postulate states that there is at least one number P_n + k in (P_n, 2P_n) such that P_n + k is a prime. Let P_n + k be the first prime P_n+1 in (P_n, 2P_n). Then Bertrand’s postulate was the first major theorem showing

$P_n{}_{+1}<2P_n$ , (2.1)

which yields the bound

$g_n=P_n{}_{+1}-P_n<P_n$ . (2.2)

It implies that growth of primes is linearly controlled. The result was historically the first genuine control on prime gaps and also one of the earliest important results in the study of primes in intervals. Bertrand theorem actually controls primes in (x, x + x), that is, interval length is x to be linear relative size. The theorem implies prime gaps cannot grow too large and provides a fundamental lower bound on prime density. Although Bertrand theorem is not yet a short interval result in the modern sense, the theorem is the starting point of the entire short interval theory because almost all later developments ask: Can the interval length x be reduced to x^θ for θ < 1? A research result is that there is at least one prime in (x, x + x^0.547) [20] and the recent development is that there is at least one prime in (x, x + x^0.525) [21]. In this paper, we only consider Bertrand theorem shows an important result such that gap between primes is linearly controlled and such linear control may always be more easily handled than nonlinear control in technique.

2.2. Bertrand Theorem and the Infinitude of Primes

Since Euclid proved infinitude of primes using constructive method [22], there are over a hundred distinct proofs for the infinitude of primes across different areas of mathematics. Of these methods, Euclid’s proof by using constructive method is the oldest, the simplest and the most perfect method as the following statement does. Suppose P_n is the largest prime. Then P₁∙ P₂∙ P₃∙…∙P_n + 1 has a prime divisor not among the assumed finite list. Thus, there is no the largest prime and primes are infinite. Representative methods to prove the infinitude of primes are as follows. Euler proved the infinitude of primes using analytic method in 1737 [23]. Dirichlet proved the infinitude of primes using arithmetic progressions in 1837 [24]. Erdös proved the infinitude of primes using elementary method in 1932 [19]. Furstenberg proved the infinitude of primes using topological method in 1955 [25]. Elsholtz proved the infinitude of primes using algebraic and combinatorial method in 2009 [26]. Meštrović proved the infinitude of primes using modern very short proofs in 2017 [27]. It should be emphasized that the prime number theorem is a very strong method to prove the infinitude of primes and one must ask: Can Bertrand theorem provide an independent proof for infinitude of primes? As we know, Bertrand’s postulate itself does not contain infinitude of primes, Chebyshev’s proof, Ramanujan’s proof and Erdös’s proof also do not contain infinitude of primes. Thus, if Bertrand’s postulate can provide a proof for infinitude of primes then the proof is independent and reliable. By Bertrand theorem, there is the following proof. Suppose P_n is the largest prime. Then there is a prime P_n+1 greater than P_n such that P_n < P_n+1 < 2P_n by Bertrand theorem. Thus, there is no the largest prime and primes are infinite. It is similar to Euclid’s proof and seems to be simpler than Euclid’s method, however, three proofs of Bertrand theorem are more complex than definition of prime. Thus, one can conclude the proof for infinitude of primes given by Bertrand theorem is independent and reliable.

2.3. Generalization of Bertrand Theorem by Ramanujan Prime

Let P_n denote the n-th prime. Then natural numbers {x} form a dense sequence and the prime sequence {P_n} can be viewed as a “compressed natural number axis”. Although the index n itself is just a natural number, it encodes the existence of the n-th prime. Therefore, one is naturally led to ask: Can Bertrand theorem be generalized from intervals on the natural number axis (x, 2x) to structures indexed by primes? Historically, mathematicians did move in this direction, and the most important development is the theory of Ramanujan primes. As is known, Ramanujan primes can be viewed as the most important and natural higher-order version of Bertrand theorem, which is historically the most famous generalization. The central idea is: Instead of merely asking whether the interval (x/2, x) contains at least one prime, one wants to know how many primes can be contained in (x/2, x). It transforms Bertrand theorem from a mere existence problem into a local density problem. Let π(x) be prime counting function and denote the number of primes not greater than x. Then Bertrand theorem is equivalent to

$\pi \left(x\right)-\pi \left(x/2\right)\ge 1$ for all $x\ge 2$ . (2.3)

When reproving Bertrand’s postulate in 1919 [18], Ramanujan asked: Can one require that the interval (x/2, x) contains not just at least one prime but at least n primes? It leads to the definition of the n-th Ramanujan prime R_n such that

$\pi \left(x\right)-\pi \left(x/2\right)\ge n$ for all $x\ge R_n$ . (2.4)

Since R₁ = 2, Bertrand theorem becomes the first Ramanujan prime case. The first five Ramanujan primes are 2, 11, 17, 29, 41 to correspond to n = 1, 2, 3, 4, 5. Later Sondow systematically studied Ramanujan primes and got the core result such that R_n ~ P_2n [28].

From above statements we see that definition of Ramanujan prime is yet established on interval of natural numbers (x/2, x) but π(x) − π(x/2) ≥ 1 for all x ≥ 2 has been strengthened as π(x) − π(x/2) ≥ n for all x ≥ R_n. It successfully transforms Bertrand theorem from a mere existence problem into a local density problem. Note that the second order natural number sequence {n} to encode the existence of primes P_n did not be specially considered in Ramanujan’s generalization of Bertrand theorem. Therefore, we feel there may be another research direction to generalize Bertrand theorem: If we directly generalize Bertrand theorem arising from the natural number sequence {x} into the second order natural number sequence {n}, then we will structure a new mathematical frame in interval (n, 2n) on the second order natural number axis. It would not strengthen Bertrand theorem but can establish the Bertrand-type theorem or conjecture so that some open problems such as Goldbach conjecture and twin prime conjecture may be better studied.

2.4. Prime Number Theorem for Double Prime

Definition 2.1. Let P_n denote the n-th prime. Then P_n is called a double prime if n is a prime p, that is, P_n = P_p.

The first 50 double primes are listed as follows

P₂, P₃, P₅, P₇, P₁₁, P₁₃, P₁₇, P₁₉, P₂₃, P₂₉, P₃₁, P₃₇, P₄₁, P₄₃, P₄₇, P₅₃, P₅₉, P₆₁, P₆₇, P₇₁, P₇₃, P₇₉, P₈₃, P₈₉, P₉₇, P₁₀₁, P₁₀₃, P₁₀₇, P₁₀₉, P₁₁₃, P₁₂₇, P₁₃₁, P₁₃₇, P₁₃₉, P₁₄₉, P₁₅₁, P₁₅₇, P₁₆₃, P₁₆₇, P₁₇₃, P₁₇₉, P₁₈₁, P₁₉₁, P₁₉₃, P₁₉₇, P₁₉₉, P₂₁₁, P₂₂₃, P₂₂₇, P₂₂₉.

It is obvious that the prime index sequence {n} is the second order natural number sequence to encode the existence of the prime sequence {P_n}. First, the sequence {n} itself can be viewed as a natural number sequence and some research results on primes among natural numbers are yet effective. Second, these results must be explained again using n to be a second order natural number encoding the existence of the n-th prime. In order to give a clear description for gap between indexes of primes, we have further definition.

Definition 2.2. Number n is called root of prime and also a prime root if P_n denotes the n-th prime.

By Definition 2.2, if P_n is a double prime P_p then n = p is a double prime root.

Definition 2.3. Let P_A(i) denote the i-th double prime. Then g_A(i) is called the i-th double prime root gap if g_A(i) = A(i + 1) − A(i).

The first 50 double prime root gaps are listed as follows

1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4.

Definition 2.4 Let P_A(i) denote the i-th double prime. Then $g_{A\left(i\right)}^{doub}$ is called the i-th double prime gap if $g_{A\left(i\right)}^{doub}$ = P_A(i+1) − P_A(i).

The first 50 double prime gaps are listed as follows

2, 6, 6, 14, 10, 18, 8, 16, 26, 18, 30, 22, 12, 20, 30, 36, 6, 48, 22, 14, 34, 30, 30, 48, 38, 16, 24, 12, 18, 92, 30, 34, 24, 62, 18, 42, 48, 24, 40, 32, 24, 66, 18, 30, 16, 80, 112, 24,14, 24.

Note that double prime is known as prime-indexed prime (PIP) in many studies. Let q_n denote prime-indexed prime P_p with p = p_n. Then it is shown that q_n ~ n(log n)² using prime-indexed prime number theorem by Barnett-Boughan [29]. Its mathematical significance is: PIPs still obey highly regular PNT-type asymptotic form. Further, higher-order prime structures were also studied. Its central asymptotic law is: $P_n^{\left(k\right)}~n{\left(\log n\right)}^k$ , corresponding to density: ~ 1/(logx)^k [30]. Thus, $P_n^{\left(2\right)}=q_n$ which is prime-indexed prime, corresponding to density: ~1/(logx)². Although such deep studies on prime-indexed primes led to many developments, in this paper we only consider existence problem of double primes among primes and prime root is an universal and useful concept which can help us to express many key mathematical objects such as gap between prime roots..

Let π(n) denote the counted number of double primes among the first n primes. Then we have the following approximation.

$\pi \left(n\right)\approx Li\left(n\right)$ , (2.5)

where

$Li\left(n\right)={\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{\log t}}$ (2.6)

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

$Li\left(n\right)\approx \frac{n}{\log n}{\displaystyle \sum _{k=0}^{\infty }\frac{k!}{{\left(\log n\right)}^k}}$ . (2.7)

Let n/log n denote the weakest form of Li(n). Then Table 1 gives counted number and predicted number by n/log n for double primes among the first n primes.

Table 1. Counted and predicted numbers for double primes among primes.

n	counted number	predicted number	π(n) − n/log n	relative error
102	25	22	3	0.1200
103	168	145	23	0.1369
104	1229	1086	143	0.1163
105	9592	8686	906	0.0944
106	78498	72382	6116	0.0779
107	664579	620421	44185	0.0664
108	5761455	5428681	332774	0.0577
109	50847534	48254942	2592592	0.0509

Theorem 2.5. Let π(n) denote counted number of double primes among primes and D^doub(n) denote average density of double primes among primes.

If $\underset{n\to \infty }{\lim }\frac{\pi \left(n\right)}{{\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{\log t}}}=1$ , then $D^{doub}\left(n\right)~\frac{1}{\log n}$ .

Proof. Using (2.7), Li(n) can be written as

$\frac{n}{\log n}{\displaystyle \sum _{k=0}^{\infty }\frac{k!}{{\left(\log n\right)}^k}}$ . (2.8)

Considering asymptotic series (2.8), 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.

$\underset{n\to \infty }{\lim }\frac{\frac{\left(k+1\right)!n}{{\left(\log n\right)}^{k+2}}}{\frac{k!n}{{\left(\log n\right)}^{k+1}}}=\underset{n\to \infty }{\lim }\frac{k+1}{\log n}=0$ . (2.9)

Since it is assumed that

$\underset{n\to \infty }{\lim }\frac{\pi \left(n\right)}{{\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{\log t}}}=1$ ,

by (2.9) we have

$\underset{n\to \infty }{\lim }\frac{\pi \left(n\right)}{\left(\frac{n}{\log n}\right)}=1$ . (2.10)

The limit (2.10) means that

$\pi \left(n\right)~\frac{n}{\log n}$ . (2.11)

Since $D^{doub}\left(n\right)=\frac{\pi \left(n\right)}{n}$ , we obtain

$D^{doub}\left(n\right)~\frac{1}{\log n}$ . (2.12)

Hence the theorem holds.

Corollary 2.6. If $\underset{n\to \infty }{\lim }\frac{\pi \left(n\right)}{{\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{\log t}}}=1$ , then there are infinitely many double primes.

Proof. By Theorem 2.5, if $\underset{n\to \infty }{\lim }\frac{\pi \left(n\right)}{{\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{\log t}}}=1$

then we have

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

Since n/log n approaches infinity as n grows without bound, π(n) approaches infinity as n grows without bound. It means there are infinitely many double primes among primes, that is, there are infinitely many double primes. Hence the corollary holds.

Remark 2.7 When every second order natural number n is viewed as a natural number, the prime number theorem for expressing asymptotic distribution law of primes among natural numbers can be used for expressing asymptotic distribution law of double primes among primes. Thus, the prime number theorem has become the prime number theorem for double prime if {n} is considered as the second order natural number axis to generate double primes, that is, asymptotic expression (2.11) can be called prime number theorem for double prime. So, all above results can be thought as results which had been proven.

2.5. Bertrand-Type Theorem for Double Prime

We can generalize Bertrand theorem into the prime root sequence {n} and have the following Bertrand-type theorem for double prime.

Theorem 2.8. Let n denote root of P_n and 2n denote root of P_2n. Then there is at least one prime root n + k in (n, 2n) for n > 1 such that n + k is a double prime root, correspondingly, there is at least one prime P_n+k in (P_n, P_2n) for n > 1 such that P_n+k is a double prime.

Remark 2.9. The front part of Theorem 2.8 is a restatement of Bertrand’s postulate on n-axis. If the prime root sequence {n} is merely viewed as a natural number sequence, Chebyshev’s proof, Ramanujan’s proof and Erdös’s proof are naturally reasonable proofs for the front part of the theorem. The latter part of Theorem 2.8 is a restatement for the front part by definition of prime root, thus, two parts of Theorem 2.8 are equivalent and the restatement naturally holds.

Corollary 2.10. Let A(i) denote the i-th double prime root. Then there are A(i + 1) < 2A(i) and A(i + 1) − A(i) < A(i).

Proof. Taking n = A(i), by Theorem 2.8, there is at least one prime root A(i) + k in (A(i), 2A(i)) such that A(i) + k is a double prime root. Let A(i) + k = A(i + 1) be the first double prime root in (A(i), 2A(i)). Then we have A(i) < A(i + 1) < 2A(i), that is, A(i + 1) < 2A(i) and A(i + 1) − A(i) < A(i). Hence the corollary holds.

Remark 2.11. Corollary 2.10 remains lineal control on gap between double prime roots. It is an important step for proving the infinitude of double prime roots.

Corollary 2.12. Let P_A(i) denote the i-th double prime. Then there are P_A(i+1) < P_2A(i) and P_A(i+1) − P_A(i) < P_2A(i) − P_A(i).

Proof. Theorem 2.8 states there is at least one prime P_n+k in (P_n, P_2n) for n > 1 such that P_n+k is a double prime. Take P_n = P_A(i). Let P_n+k = P_A(i+1) be the first double prime in (P_A(i), P_2A(i)). Then we have

$P_A{}_{\left(i+1\right)}<P_{2A}{}_{\left(i\right)}$ , (2.13)

equivalently,

$P_A{}_{\left(i+1\right)}-P_A{}_{\left(i\right)}<P_{2A}{}_{\left(i\right)}-P_A{}_{\left(i\right)}$ . (2.14)

Hence the corollary holds.

Remark 2.13. Corollary 2.12 means gap between double primes is nonlinearly controlled. It is different from gap between double prime roots which is linearly controlled. Note that P_2n > 2P_n for n > 1 [17], thus, P_2n − P_n > P_n for n > 1. Correspondingly, there is result such that P_2A(i) − P_A(i) > P_A(i) for i ≥ 1 and P_2A(i) − P_A(i) yields the bound on double prime gap as (2.14) shows. It is different from (2.2) which shows prime gap to be linearly controlled.

Theorem 2.14. There are infinitely many double primes.

Proof. Suppose A(i) is the largest double prime root. Then there is a double prime root A(i + 1) greater than A(i) such that A(i) < A(i + 1) < 2A(i) by Corollary 2.10. Thus, there is no the largest double prime root and double prime roots are infinite. Since every double prime root A(i) links with a double prime P_A(i), there are infinitely many double primes among primes, that is, there are infinitely many double primes. Hence the theorem holds.

Note that Theorem 2.8, Corollary 2.10 and Corollary 2.12 form a Bertrand-type system for double prime, which is merely an existence problem of double prime among primes.

2.6. Verification of Bertrand-Type Theorem for Double Prime

Verification of Bertrand-type theorem for double prime is divided into two parts: verification of Theorem 2.8 as well as verification of Corollary 2.10 and Corollary 2.12.

Table 2 verifies Theorem 2.8 using the number of double prime roots in (n, 2n) greater than 0 and the number of double primes in (P_n, P_2n) greater than 0 for 1 < n ≤ 50. Values of these numbers can be counted based on all double prime roots and all double primes given by the table. Figure 1 gives all numerical evidences to verify Theorem 2.8 for 1 < n ≤ 300,000 because data curve in the figure shows the number of double primes in (P_n, P_2n) to be greater than 0 for 1 < n ≤ 300,000, equivalently, the figure also verifies the number of double prime roots in (n, 2n) to be greater than 0 for 1 < n ≤ 300,000 because the number of double prime roots in (n, 2n) is always equal to the number of double primes in (P_n, P_2n) for n > 1. Thus, Theorem 2.8 has been verified up to n = 300,000. All raw data to show counted number of double primes in (P_n, P_2n) for every n for 1 < n ≤ 300,000, which have been drawn as curve in Figure 1, can be found in [31]. For example, the number of double primes in (P₅₂₅, P₁₀₅₀) is counted as 77 and the number of double primes in (P₉₃₆, P₁₈₇₂) is counted as 128 by raw data in [31]. The reference [31] includes 15105 pages to show raw data for double primes less than 10,000,000, which can cover any counting require for double primes up to n = 300,000.

Table 2. Double prime roots in (n, 2n) and double primes in (P_n, P_2n) for 1 < n ≤ 50.

n	all double prime roots	2n	Pn	all double primes	P2n
2	3	4	P2	P3	P4
3	5	6	P3	P5	P6
4	5, 7	8	P4	P5, P7	P8
5	7	10	P5	P7	P10
6	7, 11	12	P6	P7, P11	P12
7	11, 13	14	P7	P11, P13	P14
8	11, 13	16	P8	P11, P13	P16
9	11, 13, 17	18	P9	P11, P13, P17	P18
10	11, 13, 17, 19	20	P10	P11, P13, P17, P19	P20
11	13, 17, 19	22	P11	P13, P17, P19	P22
12	13, 17, 19, 23	24	P12	P13, P17, P19, P23	P24
13	17, 19, 23	26	P13	P17, P19, P23	P26
14	17, 19, 23	28	P14	P17, P19, P23	P28
15	17, 19, 23, 29	30	P15	P17, P19, P23, P29	P30
16	17, 19, 23, 29, 31	32	P16	P17, P19, P23, P29, P31	P32
17	19, 23, 29, 31	34	P17	P19, P23, P29, P31	P34
18	19, 23, 29, 31	36	P18	P19, P23, P29, P31	P36
19	23, 29, 31, 37	38	P19	P23, P29, P31, P37	P38
20	23, 29, 31, 37	40	P20	P23, P29, P31, P37	P40
21	23, 29, 31, 37, 41	42	P21	P23, P29, P31, P37, P41	P42
22	23, 29, 31, 37, 41, 43	44	P22	P23, P29, P31, P37, P41, P43	P44
23	29, 31, 37, 41, 43	46	P23	P29, P31, P37, P41, P43	P46
24	29, 31, 37, 41, 43, 47	48	P24	P29, P31, P37, P41, P43, P47	P48
25	29, 31, 37, 41, 43, 47	50	P25	P29, P31, P37, P41, P43, P47	P50
26	29, 31, 37, 41, 43, 47	52	P26	P29, P31, P37, P41, P43, P47	P52
27	29, 31, 37, 41, 43, 47, 53	54	P27	P29, P31, P37, P41, P43, P47, P53	P54
28	29, 31, 37, 41, 43, 47, 53	56	P28	P29, P31, P37, P41, P43, P47, P53	P56
29	31, 37, 41, 43, 47, 53	58	P29	P31, P37, P41, P43, P47, P53	P58
30	31, 37, 41, 43, 47, 53, 59	60	P30	P31, P37, P41, P43, P47, P53, P59	P60
31	37, 41, 43, 47, 53, 59, 61	62	P31	P37, P41, P43, P47, P53, P59, P61	P62
32	37, 41, 43, 47, 53, 59, 61	64	P32	P37, P41, P43, P47, P53, P59, P61	P64
33	37, 41, 43, 47, 53, 59, 61	66	P33	P37, P41, P43, P47, P53, P59, P61	P66
34	37, 41, 43, 47, 53, 59, 61, 67	68	P34	P37, P41, P43, P47, P53, P59, P61, P67	P68
35	37, 41, 43, 47, 53, 59, 61, 67	70	P35	P37, P41, P43, P47, P53, P59, P61, P67	P70
36	37, 41, 43, 47, 53, 59, 61, 67, 71	72	P36	P37, P41, P43, P47, P53, P59, P61, P67, P71	P72
37	41, 43, 47, 53, 59, 61, 67, 71, 73	74	P37	P41, P43, P47, P53, P59, P61, P67, P71, P73	P74
38	41, 43, 47, 53, 59, 61, 67, 71, 73	76	P38	P41, P43, P47, P53, P59, P61, P67, P71, P73	P76
39	41, 43, 47, 53, 59, 61, 67, 71, 73	78	P39	P41, P43, P47, P53, P59, P61, P67, P71, P73	P78
40	41, 43, 47, 53, 59, 61, 67, 71, 73, 79	80	P40	P41, P43, P47, P53, P59, P61, P67, P71, P73, P79	P80
41	43, 47, 53, 59, 61, 67, 71, 73, 79	82	P41	P43, P47, P53, P59, P61, P67, P71, P73, P79	P82
42	43, 47, 53, 59, 61, 67,71, 73, 79, 83	84	P42	P43, P47, P53, P59, P61, P67, P71, P73, P79, P83	P84
43	47, 53, 59, 61, 67, 71, 73, 79, 83	86	P43	P47, P53, P59, P61, P67, P71, P73, P79, P83	P86
44	47, 53, 59, 61, 67, 71, 73, 79, 83	88	P44	P47, P53, P59, P61, P67, P71, P73, P79, P83	P88
45	47, 53, 59, 61, 67, 71, 73, 79, 83, 89	90	P45	P47, P53, P59, P61, P67, P71, P73, P79, P83, P89	P90
46	47, 53, 59, 61, 67, 71, 73, 79, 83, 89	92	P46	P47, P53, P59, P61, P67, P71, P73, P79, P83, P89	P92
47	53, 59, 61, 67, 71, 73, 79, 83, 89	94	P47	P53, P59, P61, P67, P71, P73, P79, P83, P89	P94
48	53, 59, 61, 67, 71, 73, 79, 83, 89	96	P48	P53, P59, P61, P67, P71, P73, P79, P83, P89	P96
49	53, 59, 61, 67, 71, 73, 79, 83, 89, 97	98	P49	P53, P59, P61, P67, P71, P73, P79, P83, P89, P97	P98
50	53, 59, 61, 67, 71, 73, 79, 83, 89, 97	100	P50	P53, P59, P61, P67, P71, P73, P79, P83, P89, P97	P100

Figure 1. Counted number of double primes in (P_n, P_2n) for 1 < n ≤ 300,000.

Table 3 verifies A(i + 1) − A(i) < A(i) and P_A(i+1) − P_A(i) < P_2A(i) − P_A(i) for i ≤ 50. Because A(i + 1) − A(i) < A(i) and P_A(i+1) − P_A(i) < P_2A(i) − P_A(i) are two corollaries of Theorem 2.8 and the theorem has been verified up to n = 300,000 in Figure 1, the two corollaries have also been verified up to n = 300,000.

Table 3. Numerical evidence verifying Corollary 2.10 and Corollary 2.12 for i ≤ 50.

i	A(i)	A(i + 1) − A(i)	PA(i)	PA(i+1) − PA(i)	P2A(i) − PA(i)
1	2	3 − 2 = 1 < 2	P2 = 3	5 − 3 = 2	7 − 3 = 4
2	3	5 − 3 = 2 < 3	P3 = 5	11 − 5 = 6	13 − 5 = 8
3	5	7 − 5 = 2 < 5	P5 = 11	17 − 11 = 6	29 − 11 = 18
4	7	11 − 7 = 4 < 7	P7 = 17	31 − 17 = 14	43 − 17 = 26
5	11	13 − 11 = 2 < 11	P11 = 31	41 − 31 = 10	79 − 31 = 48
6	13	17 − 13 = 4 < 13	P13 = 41	59 − 41 = 18	101 − 41 = 60
7	17	19 − 17 = 2 < 17	P17 = 59	67 − 59 = 8	139 − 59 = 80
8	19	23 − 19 = 4 < 19	P19 = 67	83 − 67 = 16	163 − 67 = 96
9	23	29 − 23 = 6 < 23	P23 = 83	109 − 83 = 26	199 − 83 = 116
10	29	31 − 29 = 2 < 29	P29 = 109	127 − 109 = 18	271− 109 = 162
11	31	37 − 31 = 6 < 31	P31 = 127	157 − 127 = 30	293 − 127 = 166
12	37	41 − 37 = 4 < 37	P37 = 157	179 − 157 = 22	373 − 157 = 216
13	41	43 − 41 = 2 < 41	P41 = 179	191 − 179 = 12	421 − 179 = 242
14	43	47 − 43 = 4 < 43	P43 = 191	211 − 191 = 20	443 − 191 = 252
15	47	53 − 47 = 6 < 47	P47 = 211	241 − 211 = 30	491 − 211 = 280
16	53	59 − 53 = 6 < 53	P53 = 241	277 − 241 = 36	577 − 241 = 336
17	59	61 − 59 = 2 < 59	P59 = 277	283 − 277 = 6	647 − 277 = 370
18	61	67 − 61 = 6 < 61	P61 = 283	331 − 283 = 48	673 − 283 = 390
19	67	71 − 67 = 4 < 67	P67 = 331	353 − 331 = 22	757 − 331 = 426
20	71	73 − 71 = 2 < 71	P71 = 353	367 − 353 = 14	821 − 353 = 468
21	73	79 − 73 = 6 < 73	P73 = 367	401 − 367 = 34	839 − 367 = 472
22	79	83 − 79 = 4 < 79	P79 = 401	431 − 401 = 30	929 − 401 = 528
23	83	89 − 83 = 6 < 83	P83 = 431	461 − 431 = 30	983 − 431 = 552
24	89	97 − 89 = 8 < 89	P89 = 461	509 − 461 = 48	1061 − 461 = 600
25	97	101 − 97 = 8 < 97	P97 = 509	547 − 509 = 38	1181 − 509 = 672
26	101	103 − 101 = 2 < 101	P101 = 547	563 − 547 = 16	1231 − 547 = 684
27	103	107 − 103 = 4 < 103	P103 = 563	587 − 563 = 24	1277 − 563 = 714
28	107	109 − 107 = 2 < 107	P107 = 587	599 − 587 = 12	1307 − 587 = 720
29	109	113 − 109 = 4 < 109	P109 = 599	617 − 599 = 18	1361 − 599 = 762
30	113	127 − 113 = 14 < 113	P113 = 617	709 − 617 = 92	1429 − 617 = 812
31	127	131 − 127 = 4 < 127	P127 = 709	739 − 709 = 30	1609 − 709 = 900
32	131	137 − 131 = 6 < 131	P131 = 739	773 − 739 = 34	1667 − 739 = 928
33	137	139 − 137 = 2 < 137	P137 = 773	797 − 773 = 24	1759 − 773 = 986
34	139	149 − 139 = 10 < 139	P139 = 797	859 − 797 = 62	1789 − 797 = 992
35	149	151 − 149 = 2 < 149	P149 = 859	877 − 859 = 18	1973 − 859 = 1114
36	151	157 − 151 = 6 < 151	P151 = 877	919 − 877 = 42	1997 − 877 = 1120
37	157	163 − 157 = 6 < 157	P157 = 919	967 − 919 = 48	2083 − 919 = 1164
38	163	167 − 163 = 4 < 163	P163 = 967	991 − 967 = 24	2161 − 967 = 1194
39	167	173 − 167 = 6 < 167	P167 = 991	1031 − 991 = 40	2243 − 991 = 1252
40	173	179 − 173 = 6 < 173	P173 = 1031	1063 − 1031 = 32	2339 − 1031 = 1308
41	179	181 − 179 = 2 < 179	P179 = 1063	1087 − 1063 = 24	2411 − 1063 = 1348
42	181	191 − 181 = 10 < 181	P181 = 1087	1153 − 1087 = 66	2441 − 1087 = 1354
43	191	193 − 191 = 2 < 191	P191 = 1153	1171 − 1153 = 18	2633 − 1153 = 1480
44	193	197 − 193 = 4 < 193	P193 = 1171	1201 − 1171 = 30	2663 − 1171 = 1492
45	197	199 − 197 = 2 < 197	P197 = 1201	1217 − 1201 = 16	2707 − 1201 = 1506
46	199	211 − 199 = 12 < 199	P199 = 1217	1297 − 1217 = 80	2729 − 1217 = 1512
47	211	223 − 211 = 12 < 211	P211 = 1297	1409 − 1297 = 112	2917 − 1297 = 1620
48	223	227 − 223 = 4 < 223	P223 = 1409	1433 − 1409 = 24	3137 − 1409 = 1728
49	227	229 − 227 = 2 < 227	P227 = 1433	1447 − 1433 = 14	3209 − 1433 = 1776
50	229	233 − 229 = 4 < 229	P229 = 1447	1471 − 1447 = 24	3251 − 1447 = 1804

2.7. A Standard Model for Generalizing Bertrand Theorem

All results of the Bertrand-type theorem for double prime are proved results and form a standard model for generalizing Bertrand theorem into other kinds of second order primes. There are three steps to be suitable for the model. Step 1. Set an enough large prime range to compare counted number of a kind of second order primes with counted number of double primes. We set the range is n = 10⁹, that is, the first 1,000,000,000 primes. Step 2. If counted number of a kind of second order primes is greater than π(n) for n = 10⁹ then a Bertrand-Type Conjecture for the kind of second order primes can be proposed and corresponding corollaries can be established. Step 3. Verify the Bertrand-type conjecture and its corollaries up to n = 300,000 and the verification range accords with Bertrand’s verification range for his conjecture (up to x = 3,000,000) because x expresses existence of natural number but n expresses existence of prime. If there is no counterexample in verifying results then the Bertrand-type conjecture may be true but it requires a rigorous proof. We discover that Goldbach conjecture, which can be transformed into a problem about infinitude of a kind of special primes, and twin prime conjecture are suitable for the standard model, thus, we have the following discussions.

3. Bertrand-type Conjecture for Nontrivial Prime

3.1. The Largest Strong Goldbach Number and Nontrivial Prime

Definition 3.1. An even number L_n is called the largest strong Goldbach number generated by the n-th prime P_n if

$L_n=\max \left\{m:\left\{4,6,\cdots ,m\right\}\subset S_n\right\},S_n=\left\{P_i+P_j:P_i,P_j\le P_n\right\}$ . (3.1)

By Definition 3.1, 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 can be found in [31].

Remark 3.2. Definition 3.1 means 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. Thus, L_n ≤ L_n+1 for all n. There must exist growth points of L_n. If L_n−1 < L_n, then n is a growth point of L_n and corresponding P_n is called a nontrivial prime as the following definition.

Definition 3.3. P_n is called a nontrivial prime if P_n satisfies

$L_n{}_{–1}+2\in \left\{P_i+P_n:P_i\le P_n\right\}$ . (3.2)

By Definition 3.3, the first 50 nontrivial primes are listed as follows

P₂, P₃, P₄, P₅, P₆, P₇, P₈, P₉, P₁₁, P₁₂ P₁₃, P₁₅, P₁₈, P₁₉, P₂₁, P₂₃, P₂₄, P₂₅, P₂₆, P₂₇, P₂₉, P₃₀, P₃₁, P₃₂, P₃₄, P₃₅, P₃₆, P₃₈, P₄₁, P₄₂, P₄₄, P₄₇, P₄₉, P₅₁, P₅₂, P₅₄, P₅₉, P₆₁, P₆₃, P₆₄, P₆₇, P₆₉, P₇₀, P₇₁, P₇₂, P₇₇, P₈₁, P₈₃, P₈₆, P₈₈.

More nontrivial primes can be found in [31].

By Definition 2.2, the first 50 nontrivial prime root gaps are listed as follows

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.

More nontrivial prime root gaps can be found in [31].

The first 50 nontrivial prime gaps are listed as follows

2, 2, 4, 2, 4, 2, 4, 8, 6, 4, 6, 14, 6, 6, 10, 6, 8, 4, 2, 6, 4, 14, 4, 8, 10, 2, 12, 16, 2, 12, 18, 16, 6, 6, 12, 26, 6, 24, 4, 20, 16, 2, 4, 6, 30, 30, 12, 12, 14, 10.

More nontrivial prime gaps can be found in [31].

Theorem 3.4. If there are infinitely many nontrivial primes, then Goldbach conjecture is true.

Proof. By Definition 3.3, every nontrivial prime constructs a growth of L_n. It means that infinitude of nontrivial primes implies L_n approaches infinity as n grows without bound. Since Goldbach conjecture is equivalent to the result that L_n approaches infinity. Hence, if there are infinitely many nontrivial primes then Goldbach conjecture is true and the theorem holds.

Remark 3.5. Definition 3.1, Definition 3.3 and Theorem 3.4 have transformed Goldbach conjecture into a problem about infinitude of a kind of special primes. It has become a foundation, on which it seems to be possible to prove Goldbach conjecture by the Bertrand-type conjecture for nontrivial prime.

3.2. PNT-Type Conjecture System for Nontrivial Prime

Let Ω(n) denote counted number of nontrivial primes among the first n primes. Then we suppose there is an approximation for Ω(n) as follows

$\Omega \left(n\right)\approx Li\left(n\right)+\frac{n}{\log n}\left(\frac{1}{\log \log n}+\frac{1}{4{\left(\log \log n\right)}^2}\right)$ , (3.3)

where

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

is the logarithmic integral and it has an equivalent form

$Li\left(n\right)\approx \frac{n}{\log n}{\displaystyle \sum _{k=0}^{\infty }\frac{k!}{{\left(\log n\right)}^k}}$ .

Take the weakest form of (3.3). Then we have

$\Omega \left(n\right)\approx \frac{n}{\log n}\left(1+\frac{1}{\log \log n}+\frac{1}{4{\left(\log \log n\right)}^2}\right)$ . (3.4)

Table 4 gives counted number of nontrivial primes and predicted number of nontrivial primes by (3.4), which shows the weakest form is a good approximation.

Proposition 3.6. Let Ω(n) denote counted number of nontrivial primes among the first n primes and D^nont(n) denote average density of nontrivial primes among primes.

If $\underset{n\to \infty }{\lim }\frac{\Omega \left(n\right)}{{\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{\log t}+\frac{n}{\log n}\left(\frac{1}{\log \log n}+\frac{1}{4{\left(\log \log n\right)}^2}\right)}}=1$ , then $D^{nont}\left(n\right)~\frac{1}{\log n}$ .

Proof. Using (3.3), Li(n) can be written as

$\frac{n}{\log n}{\displaystyle \sum _{k=0}^{\infty }\frac{k!}{{\left(\log n\right)}^k}}$ . (3.5)

Considering asymptotic series (3.5), 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.

$\underset{n\to \infty }{\lim }\frac{\frac{\left(k+1\right)!n}{{\left(\log n\right)}^{k+2}}}{\frac{k!n}{{\left(\log n\right)}^{k+1}}}=\underset{n\to \infty }{\lim }\frac{k+1}{\log n}=0$ . (3.6)

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

$\underset{n\to \infty }{\lim }\frac{\frac{n}{\log n}\frac{1}{\log \log n}}{\frac{n}{\log n}}=\underset{n\to \infty }{\lim }\frac{1}{\log \log n}=0$ , (3.7)

$\underset{n\to \infty }{\lim }\frac{\frac{n}{\log n}\frac{1}{4{\left(\log \log n\right)}^2}}{\frac{n}{\log n}}=\underset{n\to \infty }{\lim }\frac{1}{4{\left(\log \log n\right)}^2}=0$ . (3.8)

Since it is assumed that

$\underset{n\to \infty }{\lim }\frac{\Omega \left(n\right)}{{\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{\log t}+\frac{n}{\log n}\left(\frac{1}{\log \log n}+\frac{1}{4{\left(\log \log n\right)}^2}\right)}}=1$ ,

by (3.6), (3.7) and (3.8) we have

$\underset{n\to \infty }{\lim }\frac{\Omega \left(n\right)}{\left(\frac{n}{\log n}\right)}=1$ . (3.9)

The limit (3.9) means that

$\Omega \left(n\right)~\frac{n}{\log n}$ . (3.10)

Since $D^{nont}\left(n\right)=\frac{\Omega \left(n\right)}{n}$ , we obtain

$D^{nont}\left(n\right)~\frac{1}{\log n}$ . (3.11)

Hence the proposition holds.

Corollary 3.7. If $\underset{n\to \infty }{\lim }\frac{\Omega \left(n\right)}{{\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{\log t}+\frac{n}{\log n}\left(\frac{1}{\log \log n}+\frac{1}{4{\left(\log \log n\right)}^2}\right)}}=1$ , then there are infinitely many nontrivial primes and Goldbach conjecture is true.

Proof. By Proposition 3.6, if $\underset{n\to \infty }{\lim }\frac{\Omega \left(n\right)}{{\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{\log t}+\frac{n}{\log n}\left(\frac{1}{\log \log n}+\frac{1}{4{\left(\log \log n\right)}^2}\right)}}=1$

then we have

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

Since n/log n approaches infinity as n grows without bound, Ω(n) approaches infinity as n grows without bound. It means there are infinitely many nontrivial primes among primes. By Theorem 3.4, Goldbach conjecture is true. Hence the corollary holds.

Table 4. Comparison between counted and predicted numbers of nontrivial primes.

n	counted number	predicted number	relative error
100	54	38	0.2962
1000	276	229	0.1702
10000	1867	1628	0.1280
100000	13692	12594	0.0801
1000000	109564	102493	0.0645
10000000	912223	863005	0.0539
100000000	7819294	7448150	0.0474
1000000000	68459493	65433701	0.0441

Remark 3.8. Proposition 3.6 and Corollary 3.7 form a PNT-type conjecture system, which is a strong method to show infinitude of nontrivial primes for proving Goldbach conjecture. However, there is another method to imply Goldbach conjecture because counted number of nontrivial primes accords with standard model for generalizing Bertrand theorem into nontrivial primes and we have the following discussion.

3.3. Bertrand-Type Conjecture for Nontrivial Prime

Comparing Table 4 with Table 1, counted number of nontrivial primes to be 68,459,493 is greater than counted number of double primes to be 50,847,534 for n = 10⁹. It accords with Step 1 in standard model. So, we make the following conjecture by Step 2 in the model.

Conjecture 3.9. Let n denote root of P_n and 2n denote root of P_2n. Then there is at least one prime root n + k in (n, 2n) for n > 1 such that n + k is a nontrivial prime root, correspondingly, there is at least one prime P_n+k in (P_n, P_2n) for n > 1 such that P_n+k is a nontrivial prime.

Remark 3.10. Conjecture 3.9 is Bertrand-type conjecture for nontrivial prime and is a new unproved hypothesis. This conjecture will lead to the infinitude of nontrivial primes as an existence problem of nontrivial prime on n-axis. It means the number of nontrivial prime roots in (n, 2n) is equal to the number of nontrivial primes in (P_n, P_2n) for n > 1.

Corollary 3.11. Let B(i) denote the i-th nontrivial prime root. Then there are B(i + 1) < 2B(i) and B(i + 1) − B(i) < B(i).

Proof. Taking n = B(i), by Conjecture 3.9, there is at least one prime root B(i) + k in (B(i), 2B(i)) such that B(i) + k is a nontrivial prime root. Let B(i) + k = B(i + 1) be the first nontrivial prime root in (B(i), 2B(i)). Then we have B(i) < B(i + 1) < 2B(i), that is, B(i + 1) < 2B(i) and B(i + 1) − B(i) < B(i). Hence the corollary holds.

Corollary 3.12. Let P_B(i) denote the i-th nontrivial prime. Then there are P_B(i+1) < P_2B(i) and P_B(i+1) − P_B(i) < P_2B(i) − P_B(i).

Proof. Conjecture 3.9 states there is at least one prime P_n+k in (P_n, P_2n) for n > 1 such that P_n+k is a nontrivial prime. Take P_n = P_B(i). Let P_n+k = P_B(i+1) be the first nontrivial prime in (P_B(i), P_2B(i)). Then we have

$P_B{}_{\left(i+1\right)}<P_{2B}{}_{\left(i\right)}$ , (3.12)

equivalently,

$P_B{}_{\left(i+1\right)}-P_B{}_{\left(i\right)}<P_{2B}{}_{\left(i\right)}-P_B{}_{\left(i\right)}$ . (3.13)

Hence the corollary holds.

Corollary 3.13. There are infinitely many nontrivial primes among primes.

Proof. Suppose B(i) is the largest nontrivial prime root. Then there is a nontrivial prime root B(i + 1) greater than B(i) such that B(i) < B(i + 1) < 2B(i) by Corollary 3.11. Thus, there is no the largest nontrivial prime root and nontrivial prime roots are infinite. Since every nontrivial prime root B(i) links with a nontrivial prime P_B(i), there are infinitely many nontrivial primes among primes. Hence the corollary holds.

Corollary 3.14. Goldbach conjecture is true.

Proof. If Conjecture 3.9 is true, then, by Corollary 3.13 there are infinitely many nontrivial primes. By Theorem 3.4, Goldbach conjecture is true. Hence the corollary holds.

Remark 3.15. Corollary 3.14 means Goldbach conjecture may be proven as an existence problem of nontrivial primes.

All above results form a system of the Bertrand-type conjecture for nontrivial prime. The central content is: First, there is at least one prime root n + k in (n, 2n) for n > 1 such that n + k is a nontrivial prime root, equivalently, there is at least one prime P_n+k in (P_n, P_2n) for n > 1 such that P_n+k is a nontrivial prime. Second, there are B(i + 1) < 2B(i) and B(i + 1) − B(i) < B(i). Third, there are P_B(i+1) < P_2B(i), P_B(i+1) − P_B(i) < P_2B(i) − P_B(i) and P_2B(i) − P_B(i) > P_B(i) for i ≥ 1.

3.4. Verification of Bertrand-Type Conjecture for Nontrivial Prime

Verification of Bertrand-type conjecture for nontrivial prime is divided into two parts: verification of Conjecture 3.9 as well as verification of Corollary 3.11 and Corollary 3.12.

Table 5 verifies Conjecture 3.9 for 1 < n ≤ 50 using the number of nontrivial prime roots in (n, 2n) greater than 0 and the number of nontrivial primes in (P_n, P_2n) greater than 0. Values of these numbers can be counted based on all nontrivial prime roots and all nontrivial primes given by the table. Figure 2 gives all numerical evidences to verify Conjecture 3.9 for 1 < n ≤ 300,000 because data curve in the figure shows the number of nontrivial primes in (P_n, P_2n) to be greater than 0 for 1 < n ≤ 300,000, equivalently, the figure also verifies the number of nontrivial prime roots in (n, 2n) to be greater than 0 for 1 < n ≤ 300,000 because the number of nontrivial prime roots in (n, 2n) is always equal to the number of nontrivial primes in (P_n, P_2n). Thus, Conjecture 3.9 has been verified up to n = 300,000. All raw data to show counted number of nontrivial primes in (P_n, P_2n) for every n for 1 < n ≤ 300,000, which have been drawn as curve in Figure 2, can be found in [31]. For example, the number of nontrivial primes in (P₄₉₁, P₉₈₂) is counted as 106 and the number of nontrivial primes in (P₉₀₁, P₁₈₀₂) is counted as 180 by raw data in [31]. The reference [31] includes 15105 pages to show raw data for nontrivial primes less than 10,000,000, which can cover any counting require for nontrivial primes up to n = 300,000.

Table 5. Nontrivial prime roots in (n, 2n) and nontrivial primes in (P_n, P_2n).

n	all nontrivial prime roots	2n	Pn	all nontrivial primes	P2n
2	3	4	P2	P3	P4
3	4, 5	6	P3	P4, P5	P6
4	5, 6, 7	8	P4	P5, P6, P7	P8
5	6, 7, 8, 9	10	P5	P6, P7, P8, P9	P10
6	7, 8, 9, 11	12	P6	P7, P8, P9, P11	P12
7	8, 9, 11, 12, 13	14	P7	P8, P9, P11, P12, P13	P14
8	9, 11, 12, 13, 15	16	P8	P9, P11, P12, P13, P15	P16
9	11, 12, 13, 15	18	P9	P11, P12, P13, P15	P18
10	11, 12, 13, 15, 18, 19	20	P10	P11, P12, P13, P15, P18, P19	P20
11	12, 13, 15, 18, 19, 21	22	P11	P12, P13, P15, P18, P19, P21	P22
12	13, 15, 18, 19, 21, 23	24	P12	P13, P15, P18, P19, P21, P23	P24
13	15, 18, 19, 21, 23, 24, 25	26	P13	P15, P18, P19, P21, P23, P24, P25	P26
14	15, 18, 19, 21, 23, 24, 25, 26, 27	28	P14	P15, P18, P19, P21, P23, P24, P25, P26, P27	P28
15	18, 19, 21, 23, 24, 25, 26, 27, 29	30	P15	P18, P19, P21, P23, P24, P25, P26, P27, P29	P30
16	18, 19, 21, 23, 24, 25, 26, 27, 29, 30, 31	32	P16	P18, P19, P21, P23, P24, P25, P26, P27, P29, P30, P31	P32
17	18, 19, 21, 23, 24, 25, 26, 27, 29, 30, 31, 32	34	P17	P18, P19, P21, P23, P24, P25, P26, P27, P29, P30, P31, P32	P34
18	19, 21, 23, 24, 25, 26, 27, 29, 30, 31, 32, 34, 35	36	P18	P19, P21, P23, P24, P25, P26, P27, P29, P30, P31, P32, P34, P35	P36
19	21, 23, 24, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36	38	P19	P21, P23, P24, P25, P26, P27, P29, P30, P31, P32, P34, P35, P36	P38
20	21, 23, 24, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 38	40	P20	P21, P23, P24, P25, P26, P27, P29, P30, P31, P32, P34, P35, P36, P38	P40
21	23, 24, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 38, 41	42	P21	P23, P24, P25, P26, P27, P29, P30, P31, P32, P34, P35, P36, P38, P41	P42
22	23, 24, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 38, 41, 42	44	P22	P23, P24, P25, P26, P27, P29, P30, P31, P32, P34, P35, P36, P38, P41, P42	P44
23	24, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 38, 41, 42, 44	46	P23	P24, P25, P26, P27, P29, P30, P31, P32, P34, P35, P36, P38, P41, P42, P44	P46
24	25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 38, 41, 42, 44, 47	48	P24	P25, P26, P27, P29, P30, P31, P32, P34, P35, P36, P38, P41, P42, P44, P47	P48
25	26, 27, 29, 30, 31, 32, 34, 35, 36, 38, 41, 42, 44, 47, 49	50	P25	P26, P27, P29, P30, P31, P32, P34, P35, P36, P38, P41, P42, P44, P47, P49	P50
26	27, 29, 30, 31, 32, 34, 35, 36, 38, 41, 42, 44, 47, 49, 51	52	P26	P27, P29, P30, P31, P32, P34, P35, P36, P38, P41, P42, P44, P47, P49, P51	P52
27	29, 30, 31, 32, 34, 35, 36, 38, 41, 42, 44, 47, 49, 51, 52	54	P27	P29, P30, P31, P32, P34, P35, P36, P38, P41, P42, P44, P47, P49, P51, P52	P54
28	29, 30, 31, 32, 34, 35, 36, 38, 41, 42, 44, 47, 49, 51, 52, 54	56	P28	P29, P30, P31, P32, P34, P35, P36, P38, P41, P42, P44, P47, P49, P51, P52, P54	P56
29	30, 31, 32, 34, 35, 36, 38, 41, 42, 44, 47, 49, 51, 52, 54	58	P29	P30, P31, P32, P34, P35, P36, P38, P41, P42, P44, P47, P49, P51, P52, P54	P58
30	31, 32, 34, 35, 36, 38, 41, 42, 44, 47, 49, 51, 52, 54, 59	60	P30	P31, P32, P34, P35, P36, P38, P41, P42, P44, P47, P49, P51, P52, P54, P59	P60
31	32, 34, 35, 36, 38, 41, 42, 44, 47, 49, 51, 52, 54, 59, 61	62	P31	P32, P34, P35, P36, P38, P41, P42, P44, P47, P49, P51, P52, P54, P59, P61	P62
32	34, 35, 36, 38, 41, 42, 44, 47, 49, 51, 52, 54, 59, 61, 63	64	P32	P34, P35, P36, P38, P41, P42, P44, P47, P49, P51, P52, P54, P59, P61, P63	P64
33	34, 35, 36, 38, 41, 42, 44, 47, 49, 51, 52, 54, 59, 61, 63, 64	66	P33	P34, P35, P36, P38, P41, P42, P44, P47, P49, P51, P52, P54, P59, P61, P63, P64	P66
34	35, 36, 38, 41, 42, 44, 47, 49, 51, 52, 54, 59, 61, 63, 64, 67	68	P34	P35, P36, P38, P41, P42, P44, P47, P49, P51, P52, P54, P59, P61, P63, P64, P67	P68
35	36, 38, 41, 42, 44, 47, 49, 51, 52, 54, 59, 61, 63, 64, 67, 69	70	P35	P36, P38, P41, P42, P44, P47, P49, P51, P52, P54, P59, P61, P63, P64, P67, P69	P70
36	38, 41, 42, 44, 47, 49, 51, 52, 54, 59, 61, 63, 64, 67, 69, 70, 71	72	P36	P38, P41, P42, P44, P47, P49, P51, P52, P54, P59, P61, P63, P64, P67, P69, P70, P71	P72
37	38, 41, 42, 44, 47, 49, 51, 52, 54, 59, 61, 63, 64, 67, 69, 70, 71, 72	74	P37	P38, P41, P42, P44, P47, P49, P51, P52, P54, P59, P61, P63, P64, P67, P69, P70, P71, P72	P74
38	41, 42, 44, 47, 49, 51, 52, 54, 59, 61, 63, 64, 67, 69, 70, 71, 72	76	P38	P41, P42, P44, P47, P49, P51, P52, P54, P59, P61, P63, P64, P67, P69, P70, P71, P72	P76
39	41, 42, 44, 47, 49, 51, 52, 54, 59, 61, 63, 64, 67, 69, 70, 71, 72, 77	78	P39	P41, P42, P44, P47, P49, P51, P52, P54, P59, P61, P63, P64, P67, P69, P70, P71, P72, P77	P78
40	41, 42, 44, 47, 49, 51, 52, 54, 59, 61, 63, 64, 67, 69, 70, 71, 72, 77	80	P40	P41, P42, P44, P47, P49, P51, P52, P54, P59, P61, P63, P64, P67, P69, P70, P71, P72, P77	P80
41	42, 44, 47, 49, 51, 52, 54, 59, 61, 63, 64, 67, 69, 70, 71, 72, 77, 81	82	P41	P42, P44, P47, P49, P51, P52, P54, P59, P61, P63, P64, P67, P69, P70, P71, P72, P77, P81	P82
42	44, 47, 49, 51, 52, 54, 59, 61, 63, 64, 67, 69, 70, 71, 72, 77, 81, 83	84	P42	P44, P47, P49, P51, P52, P54, P59, P61, P63, P64, P67, P69, P70, P71, P72, P77, P81, P83	P84
43	44, 47, 49, 51, 52, 54, 59, 61, 63, 64, 67, 69, 70, 71, 72, 77, 81, 83	86	P43	P44, P47, P49, P51, P52, P54, P59, P61, P63, P64, P67, P69, P70, P71, P72, P77, P81, P83	P86
44	47, 49, 51, 52, 54, 59, 61, 63, 64, 67, 69, 70, 71, 72, 77, 81, 83, 86	88	P44	P47, P49, P51, P52, P54, P59, P61, P63, P64, P67, P69, P70, P71, P72, P77, P81, P83, P86	P88
45	47, 49, 51, 52, 54, 59, 61, 63, 64, 67, 69, 70, 71, 72, 77, 81, 83, 86, 88	90	P45	P47, P49, P51, P52, P54, P59, P61, P63, P64, P67, P69, P70, P71, P72, P77, P81, P83, P86, P88	P90
46	47, 49, 51, 52, 54, 59, 61, 63, 64, 67, 69, 70, 71, 72, 77, 81, 83, 86, 88, 91	92	P46	P47, P49, P51, P52, P54, P59, P61, P63, P64, P67, P69, P70, P71, P72, P77, P81, P83, P86, P88, P91	P92
47	49, 51, 52, 54, 59, 61, 63, 64, 67, 69, 70, 71, 72, 77, 81, 83, 86, 88, 91, 93	94	P47	P49, P51, P52, P54, P59, P61, P63, P64, P67, P69, P70, P71, P72, P77, P81, P83, P86, P88, P91, P93	P94
48	49, 51, 52, 54, 59, 61, 63, 64, 67, 69, 70, 71, 72, 77, 81, 83, 86, 88, 91, 93, 95	96	P48	P49, P51, P52, P54, P59, P61, P63, P64, P67, P69, P70, P71, P72, P77, P81, P83, P86, P88, P91, P93, P95	P96
49	51, 52, 54, 59, 61, 63, 64, 67, 69, 70, 71, 72, 77, 81, 83, 86, 88, 91, 93, 95, 96	98	P49	P51, P52, P54, P59, P61, P63, P64, P67, P69, P70, P71, P72, P77, P81, P83, P86, P88, P91, P93, P95, P96	P98
50	51, 52, 54, 59, 61, 63, 64, 67, 69, 70, 71, 72, 77, 81, 83, 86, 88, 91, 93, 95, 96	100	P50	P51, P52, P54, P59, P61, P63, P64, P67, P69, P70, P71, P72, P77, P81, P83, P86, P88, P91, P93, P95, P96	P100

Figure 2. Counted number of nontrivial primes in (P_n, P_2n) for 1 < n ≤ 300,000.

Table 6 verifies B(i + 1) − B(i) < B(i) and P_B(i+1) − P_B(i) < P_2B(i) − P_B(i) for i ≤ 50. Because B(i + 1) − B(i) < B(i) and P_B(i+1) − P_B(i) < P_2B(i) − P_B(i) are two corollaries of Conjecture 3.9 and the conjecture has been verified up to n = 300,000 in Figure 2, the two corollaries have also been verified up to n = 300,000.

Note that there is no counterexample in all verifying results up to n = 300,000 at Step 3 of our model. Thus, if Conjecture 3.9 is proven then Goldbach conjecture is true.

Table 6. Numerical evidence verifying Corollary 3.11 and Corollary 3.12 for i ≤ 50.

i	B(i)	B(i + 1) − B(i)	PB(i)	PB(i+1) − PB(i)	P2B(i) − PB(i)
1	2	3 − 2 = 1 < 2	P2 = 3	5 − 3 = 2	7 − 3 = 4
2	3	4 − 3 = 1 < 3	P3 = 5	7 − 5 = 2	13 − 5 = 8
3	4	5 − 4 = 1 < 4	P4 = 7	11 − 7 = 4	19 − 7 = 12
4	5	6 − 5 = 1 < 5	P5 = 11	13 − 11 = 2	29 − 11 = 18
5	6	7 − 6 = 1 < 6	P6 = 13	17 − 13 = 4	37 − 13 = 24
6	7	8 − 7 = 1 < 7	P7 = 17	19 − 17 = 2	43 − 17 = 26
7	8	9 − 8 = 1 < 8	P8 = 19	23 − 19 = 4	53 − 19 = 34
8	9	11 − 9 = 2 < 9	P9 = 23	31 − 23 = 8	61 − 23 = 38
9	11	12 − 11 = 1 < 11	P11 = 31	37 − 31 = 6	79 − 31 = 48
10	12	13 − 12 = 1 < 12	P12 = 37	41 − 37 = 4	89 − 37 = 52
11	13	15 − 13 = 2 < 13	P13 = 41	47 − 41 = 6	101 − 41 = 60
12	15	18 − 15 = 3 < 15	P15 = 47	61 − 47 = 14	113 − 47 = 66
13	18	19 − 18 = 1 < 18	P18 = 61	67 − 61 = 6	151 − 61 = 90
14	19	21 − 19 = 2 < 19	P19 = 67	73 − 67 = 6	163 − 67 = 96
15	21	23 − 21 = 2 < 21	P21 = 73	83 − 73 = 10	181 − 73 = 108
16	23	24 − 23 = 1 < 23	P23 = 83	89 − 83 = 6	199 − 83 = 116
17	24	25 − 24 = 1 < 24	P24 = 89	97 − 89 = 8	223 − 89 = 134
18	25	26 − 25 = 1 < 25	P25 = 97	101 − 97 = 4	229 − 97 = 132
19	26	27 − 26 = 1 < 26	P26 = 101	103 − 101 = 2	239 − 101 = 138
20	27	29 − 27 = 2 < 27	P27 = 103	109 − 103 = 6	251 − 103 = 148
21	29	30 − 29 = 1 < 29	P29 = 109	113 − 109 = 4	271 − 109 = 162
22	30	31 − 30 = 1 < 30	P30 = 113	127 − 113 = 14	281 − 113 = 168
23	31	32 − 31 = 1 < 31	P31 = 127	131 − 127 = 4	293 − 127 = 166
24	32	34 − 32 = 2 < 32	P32 = 131	139 − 131 = 8	311 − 131 = 180
25	34	35 − 34 = 1 < 34	P34 = 139	149 − 139 = 10	337 − 139 = 198
26	35	36 − 35 = 1 < 35	P35 = 149	151 − 149 = 2	349 − 149 = 200
27	36	38 − 36 = 2 < 36	P36 = 151	163 − 151 = 12	359 − 151 = 208
28	38	41 − 38 = 3 < 38	P38 = 163	179 − 163 = 16	383 − 163 = 220
29	41	42 − 41 = 1 < 41	P41 = 179	181 − 179 = 2	421 − 179 = 242
30	42	44 − 42 = 2 < 42	P42 = 181	193 − 181 = 12	433 − 181 = 252
31	44	47 − 44 = 3 < 44	P44 = 193	211 − 193 = 18	457 − 193 = 264
32	47	49 − 47 = 2 < 47	P47 = 211	227 − 211 = 16	491 − 211 = 280
33	49	51 − 49 = 2 < 49	P49 = 227	233 − 227 = 6	521 − 227 = 294
34	51	52 − 51 = 1 < 51	P51 = 233	239 − 233 = 6	557 − 233 = 324
35	52	54 − 52 = 2 < 52	P52 = 239	251 − 239 = 12	569 − 239 = 330
36	54	59 − 54 = 5 < 54	P54 = 251	277 − 251 = 26	593 − 251 = 342
37	59	61 − 59 = 2 < 59	P59 = 277	283 − 277 = 6	647 − 277 = 370
38	61	63 − 61 = 2 < 61	P61 = 283	307 − 283 = 24	673 − 283 = 390
39	63	64 − 63 = 1 < 63	P63 = 307	311 − 307 = 4	701 − 307 = 394
40	64	67 − 64 = 3 < 64	P64 = 311	331 − 311 = 20	719 − 311 = 408
41	67	69 − 67 = 2 < 67	P67 = 331	347 − 331 = 16	757 − 331 = 426
42	69	70 − 69 = 1 < 69	P69 = 347	349 − 347 = 2	787 − 347 = 440
43	70	71 − 70 = 1 < 70	P70 = 349	353 − 349 = 4	809 − 349 = 460
44	71	72 − 71 = 1 < 71	P71 = 353	359 − 353 = 6	821 − 353 = 468
45	72	77 − 72 = 5 < 72	P72 = 359	389 − 359 = 30	827 − 359 = 468
46	77	81 − 77 = 4 < 77	P77 = 389	419 − 389 = 30	887 − 389 = 498
47	81	83 − 81 = 2 < 81	P81 = 419	431 − 419 = 12	953 − 419 = 534
48	83	86 − 83 = 3 < 83	P83 = 431	443 − 431 = 12	983 − 431 = 552
49	86	88 − 86 = 2 < 86	P86 = 443	457 − 443 = 14	1021 − 443 = 578
50	88	91 − 88 = 3 < 88	P88 = 457	467 − 457 = 10	1049 − 457 = 592

4. Bertrand-Type Conjecture for Twin Prime

4.1. Twin Prime and the Twin Prime Conjecture

Definition 4.1. Let P_n denote the n-th prime. P_n is called a twin prime if P_n+1 − P_n = 2.

By Definition 4.1, the first 50 twin primes are listed as follows.

P₂, P₃, P₅, P₇, P₁₀, P₁₃, P₁₇, P₂₀, P₂₆, P₂₈, P₃₃, P₃₅, P₄₁, P₄₃, P₄₅, P₄₉, P₅₂, P₅₇, P₆₀, P₆₄, P₆₉, P₈₁, P₈₃, P₈₉, P₉₈, P₁₀₄, P₁₀₉, P₁₁₃, P₁₁₆, P₁₂₀, P₁₄₀, P₁₄₂, P₁₄₄, P₁₄₈, P₁₅₂, P₁₇₁, P₁₇₃, P₁₇₆, P₁₇₈, P₁₈₂, P₁₉₀, P₂₀₁, P₂₀₆, P₂₀₉, P₂₁₂, P₂₁₅, P₂₂₅, P₂₃₀, P₂₃₄, P₂₃₆.

By Definition 2.2, the first 50 twin prime root gaps are listed as follows.

1, 2, 2, 3, 3, 4, 3, 6, 2, 5, 2, 6, 2, 2, 4, 3, 5, 3, 4, 5, 12, 2, 6, 9, 6, 5, 4, 3, 4, 20, 2, 2, 4, 4, 19, 2, 3, 2, 4, 8, 11, 5, 3, 3, 3, 10, 5, 4, 2, 17.

The first 50 twin prime gaps are listed as follows.

2, 6, 6, 12, 12, 18, 12, 30, 6, 30, 12, 30, 12, 6, 30, 12, 30, 12, 30, 36, 72, 12, 30, 60, 48, 30, 18, 24, 18, 150, 12, 6, 30, 24, 138, 12, 18, 12, 30, 60, 78, 48, 12, 12, 18, 108, 24, 30, 6, 120.

Polignac conjecture states that there are infinitely many primes p such that p + 2k is also prime for every natural number k. Twin prime conjecture is the Polignac conjecture case for k = 1. Let π₂(x) denote counted number of primes p ≤ x such that p + 2 is also prime. Define twin prime constant C₂ as [32]

$C_2={\displaystyle \prod _{p\ge 3}\left(1-\frac{1}{{\left(p-1\right)}^2}\right)}={\displaystyle \prod _{p\ge 3}\frac{p\left(p-2\right)}{{\left(p-1\right)}^2}}\approx 0.660161815\cdots$ . (4.1)

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

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

${\pi }_2\left(x\right)~2C_2\frac{x}{{\left(\log x\right)}^2}$ , (4.2)

in the sense that the quotient of the two expressions approaches 1 as x grows without bound [33]. Let D^twin(x) denote average density of twin primes among natural numbers. Then from (4.2) we have

$D^{twin}\left(x\right)~2C_2\frac{1}{{\left(\log x\right)}^2}$ , (4.3)

The result is also a conjecture as asymptotic form (4.2) does.

4.2. PNT-Type Conjecture System for Twin Prime

Let π₂(n) denote counted number of twin primes among the first n primes. Then it can be conjectured that there is an approximation for π₂(n) as follows

${\pi }_2\left(n\right)\approx 2C_{2}Li\left(n\right)$ , (4.4)

where C₂ is twin prime constant (4.1) and

$Li\left(n\right)={\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{\log t}}\approx \frac{n}{\log n}{\displaystyle \sum _{k=0}^{\infty }\frac{k!}{{\left(\log n\right)}^k}}$ .

Take the weakest form of (4.4). Then we have

${\pi }_2\left(n\right)\approx 2C_2\frac{n}{\log n}$ . (4.5)

Using the weakest form to predict the number of twin primes among primes, Table 7 gives counted number of twin primes among primes and predicted number of twin primes among primes by (4.5).

Proposition 4.2. Let π₂(n) denote counted number of twin primes among the first n primes and D^twin(n) denote average density of twin primes among primes.

If $\underset{n\to \infty }{\lim }\frac{{\pi }_2\left(n\right)}{2C_2{\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{\log t}}}=1$ , then $D^{twin}\left(n\right)~2C_2\frac{1}{\log n}$ .

Proof. Using (4.4), 2C₂Li(n) can be written as

$2C_2\frac{n}{\log n}{\displaystyle \sum _{k=0}^{\infty }\frac{k!}{{\left(\log n\right)}^k}}$ . (4.6)

Considering asymptotic series (4.6), 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.

$\underset{n\to \infty }{\lim }\frac{2C_2\frac{\left(k+1\right)!n}{{\left(\log n\right)}^{k+2}}}{2C_2\frac{k!n}{{\left(\log n\right)}^{k+1}}}=\underset{n\to \infty }{\lim }\frac{k+1}{\log n}=0$ . (4.7)

Since it is assumed that

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

by (4.6) and (4.7) we have

$\underset{n\to \infty }{\lim }\frac{{\pi }_2\left(n\right)}{2C_2\frac{n}{\log n}}=1$ . (4.8)

The limit (4.8) means that

${\pi }_2\left(n\right)~2C_2\frac{n}{\log n}$ . (4.9)

Since $D^{twin}\left(n\right)=\frac{{\pi }_2\left(n\right)}{n}$ , we obtain

$D^{twin}\left(n\right)~2C_2\frac{1}{\log n}$ . (4.10)

Hence the proposition holds.

Table 7. Comparison between counted and predicted numbers of twin primes.

n	counted number	predicted number	relative error
100	25	28	0.1071
1000	174	191	0.0890
10000	1270	1433	0.1137
100000	10250	11468	0.1062
1000000	86027	95568	0.0998
10000000	738597	819156	0.0983
100000000	6497407	7167615	0.0935
1000000000	58047180	63712139	0.0889

Corollary 4.3. If $\underset{n\to \infty }{\lim }\frac{{\pi }_2\left(n\right)}{2C_2{\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{\log t}}}=1$ , then there are infinitely many twin primes.

Proof. By Proposition 4.2, if $\underset{n\to \infty }{\lim }\frac{{\pi }_2\left(n\right)}{2C_2{\displaystyle \underset{2}{\overset{n}{\int }}\frac{\text{d}t}{\log t}}}=1$ then we have

${\pi }_2\left(n\right)~2C_2\frac{n}{\log n}$ .

Since n/log n approaches infinity as n grows without bound, π₂(n) approaches infinity as n grows without bound. It means there are infinitely many twin primes among primes, that is, there are infinitely many twin primes. Hence the corollary holds.

Remark 4.4. Corollary 4.3 implies twin prime conjecture by PNT-type system for twin prime, which is a strong form to show the infinitude of twin primes.

4.3. Bertrand-Type Conjecture for Twin Prime

Comparing Table 7 with Table 1, counted number of twin primes to be 58047180 is greater than counted number of double primes to be 50847534 for n = 10⁹. It accords with Step 1 in standard model. So, we make the following conjecture by Step 2 in the model.

Conjecture 4.5. Let n denote root of P_n and 2n denote root of P_2n. Then there is at least one prime root n + k in (n, 2n) for n > 1 such that n + k is a twin prime root, correspondingly, there is at least one prime P_n+k in (P_n, P_2n) for n > 1 such that P_n+k is a twin prime.

Remark 4.6. Conjecture 4.5 is Bertrand-type conjecture for twin prime and is a new unproved hypothesis. This conjecture will lead to the infinitude of twin primes as an existence problem of twin prime on n-axis. It means the number of twin prime roots in (n, 2n) is equal to the number of twin primes in (P_n, P_2n) for n > 1.

Corollary 4.7. Let C(i) denote the i-th twin prime root. Then there are C(i + 1) < 2C(i) and C(i + 1) − C(i) < C(i).

Proof. Taking n = C(i), by Conjecture 4.5, there is at least one prime root C(i) + k in (C(i), 2C(i)) such that C(i) + k is a twin prime root. Let C(i) + k = C(i + 1) be the first twin prime root in (C(i), 2C(i)). Then we have C(i) < C(i + 1) < 2C(i), that is, C(i + 1) < 2C(i) and C(i + 1) − C(i) < C(i). Hence the corollary holds.

Corollary 4.8. Let P_C(i) denote the i-th twin prime. Then there are P_C(i+1) < P_2C(i) and P_C(i+1) − P_C(i) < P_2C(i) − P_C(i).

Proof. Conjecture 4.5 states there is at least one prime P_n+k in (P_n, P_2n) for n > 1 such that P_n+k is a twin prime. Take P_n = P_C(i). Let P_n+k = P_C(i+1) be the first twin prime in (P_C(i), P_2C(i)). Then we have

$P_C{}_{\left(i+1\right)}<P_{2C}{}_{\left(i\right)}$ , (4.11)

equivalently,

$P_C{}_{\left(i+1\right)}-P_C{}_{\left(i\right)}<P_{2C}{}_{\left(i\right)}-P_C{}_{\left(i\right)}$ . (4.12)

Hence the corollary holds.

Corollary 4.9. There are infinitely many twin primes.

Proof. Suppose C(i) is the largest twin prime root. Then there is a twin prime root C(i + 1) greater than C(i) such that C(i) < C(i + 1) < 2C(i) by Corollary 4.7. Thus, there is no the largest twin prime root and twin prime roots are infinite. Since every twin prime root C(i) links with a twin prime P_C(i), there are infinitely many twin primes among primes, that is, there are infinitely many twin primes. Hence the corollary holds.

All above results form a system of the Bertrand-type conjecture for twin prime. The central content is: First, there is at least one prime root n + k in (n, 2n) for n > 1 such that n + k is a twin prime root, equivalently, there is at least one prime P_n+k in (P_n, P_2n) for n > 1 such that P_n+k is a twin prime. Second, there are C(i + 1) < 2C(i) and C(i + 1) − C(i) < C(i). Third, there are P_C(i+1) < P_2C(i), P_C(i+1) − P_C(i) < P_2C(i) − P_C(i), P_2C(i) − P_C(i) > P_C(i) for i ≥ 1.

4.4. Verification of Bertrand-Type Conjecture for Twin Prime

Verification of Bertrand-type conjecture for twin prime is divided into two parts: verification of Conjecture 4.5 as well as verification of Corollary 4.7 and Corollary 4.8.

Table 8 verifies Conjecture 4.5 for 1 < n ≤ 50 using the number of twin prime roots in (n, 2n) greater than 0 and the number of twin primes in (P_n, P_2n) greater than 0. Values of these numbers can be counted based on all twin prime roots and all twin primes given by the table. Figure 3 gives all numerical evidences to verify Conjecture 4.5 for 1 < n ≤ 300,000 because data curve in the figure shows the number of twin primes in (P_n, P_2n) to be greater than 0 for 1 < n ≤ 300,000, equivalently, the figure also verifies the number of twin prime roots in (n, 2n) to be greater than 0 for 1 < n ≤ 300,000 because the number of twin prime roots in (n, 2n) is always equal to the number of twin primes in (P_n, P_2n). Thus, Conjecture 4.5 has been verified up to n = 300,000. All raw data to show counted number of twin primes in (P_n, P_2n) for every n for 1 < n ≤ 300,000, which have been drawn as curve in Figure 3, can be found in [31]. For example, the number of twin primes in (P₆₁₈, P₁₂₃₆) is counted as 89 and the number of twin primes in (P₉₉₃, P₁₉₈₆) is counted as 127 by raw data in [31]. The reference [31] includes 15105 pages to show raw data for twin primes less than 10,000,000, which can cover any counting require for twin primes up to n = 300,000.

Table 8. Twin prime roots in (n, 2n) and twin primes in (P_n, P_2n) for 1 < n ≤ 50.

n	all twin prime roots	2n	Pn	all twin primes	P2n
2	3	4	P2	P3	P4
3	5	6	P3	P5	P6
4	5, 7	8	P4	P5, P7	P8
5	7	10	P5	P7	P10
6	7, 10	12	P6	P7, P10	P12
7	10, 13	14	P7	P10, P13	P14
8	10, 13	16	P8	P10, P13	P16
9	10, 13, 17	18	P9	P10, P13, P17	P18
10	13, 17	20	P10	P13, P17	P20
11	13, 17, 20	22	P11	P13, P17, P20	P22
12	13, 17, 20	24	P12	P13, P17, P20	P24
13	17, 20	26	P13	P17, P20	P26
14	17, 20, 26	28	P14	P17, P20, P26	P28
15	17, 20, 26, 28	30	P15	P17, P20, P26, P28	P30
16	17, 20, 26, 28	32	P16	P17, P20, P26, P28	P32
17	20, 26, 28, 33	34	P17	P20, P26, P28, P33	P34
18	20, 26, 28, 33, 35	36	P18	P20, P26, P28, P33, P35	P36
19	20, 26, 28, 33, 35	38	P19	P20, P26, P28, P33, P35	P38
20	26, 28, 33, 35	40	P20	P26, P28, P33, P35	P40
21	26, 28, 33, 35, 41	42	P21	P26, P28, P33, P35, P41	P42
22	26, 28, 33, 35, 41, 43	44	P22	P26, P28, P33, P35, P41, P43	P44
23	26, 28, 33, 35, 41, 43, 45	46	P23	P26, P28, P33, P35, P41, P43, P45	P46
24	26, 28, 33, 35, 41, 43, 45	48	P24	P26, P28, P33, P35, P41, P43, P45	P48
25	26, 28, 33, 35, 41, 43, 45, 49	50	P25	P26, P28, P33, P35, P41, P43, P45, P49	P50
26	28, 33, 35, 41, 43, 45, 49	52	P26	P28, P33, P35, P41, P43, P45, P49	P52
27	28, 33, 35, 41, 43, 45, 49, 52	54	P27	P28, P33, P35, P41, P43, P45, P49, P52	P54
28	33, 35, 41, 43, 45, 49, 52	56	P28	P33, P35, P41, P43, P45, P49, P52	P56
29	33, 35, 41, 43, 45, 49, 52, 57	58	P29	P33, P35, P41, P43, P45, P49, P52, P57	P58
30	33, 35, 41, 43, 45, 49, 52, 57	60	P30	P33, P35, P41, P43, P45, P49, P52, P57	P60
31	33, 35, 41, 43, 45, 49, 52, 57, 60	62	P31	P33, P35, P41, P43, P45, P49, P52, P57, P60	P62
32	33, 35, 41, 43, 45, 49, 52, 57, 60	64	P32	P33, P35, P41, P43, P45, P49, P52, P57, P60	P64
33	35, 41, 43, 45, 49, 52, 57, 60, 64	66	P33	P35, P41, P43, P45, P49, P52, P57, P60, P64	P66
34	35, 41, 43, 45, 49, 52, 57, 60, 64	68	P34	P35, P41, P43, P45, P49, P52, P57, P60, P64	P68
35	41, 43, 45, 49, 52, 57, 60, 64, 69	70	P35	P41, P43, P45, P49, P52, P57, P60, P64, P69	P70
36	41, 43, 45, 49, 52, 57, 60, 64, 69	72	P36	P41, P43, P45, P49, P52, P57, P60, P64, P69	P72
37	41, 43, 45, 49, 52, 57, 60, 64, 69	74	P37	P41, P43, P45, P49, P52, P57, P60, P64, P69	P74
38	41, 43, 45, 49, 52, 57, 60, 64, 69	76	P38	P41, P43, P45, P49, P52, P57, P60, P64, P69	P76
39	41, 43, 45, 49, 52, 57, 60, 64, 69	78	P39	P41, P43, P45, P49, P52, P57, P60, P64, P69	P78
40	41, 43, 45, 49, 52, 57, 60, 64, 69	80	P40	P41, P43, P45, P49, P52, P57, P60, P64, P69	P80
41	43, 45, 49, 52, 57, 60, 64, 69, 81	82	P41	P43, P45, P49, P52, P57, P60, P64, P69, P81	P82
42	43, 45, 49, 52, 57, 60, 64, 69, 81, 83	84	P42	P43, P45, P49, P52, P57, P60, P64, P69, P81, P83	P84
43	45, 49, 52, 57, 60, 64, 69, 81, 83	86	P43	P45, P49, P52, P57, P60, P64, P69, P81, P83	P86
44	45, 49, 52, 57, 60, 64, 69, 81, 83	88	P44	P45, P49, P52, P57, P60, P64, P69, P81, P83	P88
45	49, 52, 57, 60, 64, 69, 81, 83, 89	90	P45	P49, P52, P57, P60, P64, P69, P81, P83, P89	P90
46	49, 52, 57, 60, 64, 69, 81, 83, 89	92	P46	P49, P52, P57, P60, P64, P69, P81, P83, P89	P92
47	49, 52, 57, 60, 64, 69, 81, 83, 89	94	P47	P49, P52, P57, P60, P64, P69, P81, P83, P89	P94
48	49, 52, 57, 60, 64, 69, 81, 83, 89	96	P48	P49, P52, P57, P60, P64, P69, P81, P83, P89	P96
49	52, 57, 60, 64, 69, 81, 83, 89	98	P49	P52, P57, P60, P64, P69, P81, P83, P89	P98
50	52, 57, 60, 64, 69, 81, 83, 89, 98	100	P50	P52, P57, P60, P64, P69, P81, P83, P89, P98	P100

Figure 3. Counted number of twin primes in (P_n, P_2n) for 1 < n ≤ 300,000.

Table 9. Numerical evidence verifying Corollary 4.7 and Corollary 4.8 for i ≤ 50.

i	C(i)	C(i + 1) − C(i)	PC(i)	PC(i+1) − PC(i)	P2C(i) − PC(i)
1	2	3 − 2 = 1 < 2	P2 = 3	5 − 3 = 2	7 − 3 = 4
2	3	5 − 3 = 2 < 3	P3 = 5	11 − 5 = 6	13 − 5 = 8
3	5	7 − 5 = 2 < 5	P5 = 11	17 −11 = 6	29 −11 = 18
4	7	10 − 7 = 3 < 7	P7 = 17	29 −17 = 12	43 −17 = 26
5	10	13 − 10 = 3 < 10	P10 = 29	41 −29 = 12	71 −29 = 42
6	13	17 − 13 = 4 < 13	P13 = 41	59 − 41 = 18	101 − 41 = 60
7	17	20 − 17 = 3 < 17	P17 = 59	71 − 59 = 12	139 − 59 = 80
8	20	26 − 20 = 6 < 20	P20 = 71	101 − 71 = 30	173 − 71 = 102
9	26	28 − 26 = 2 < 26	P26 = 101	107 − 101 = 6	239 − 101 = 138
10	28	33 − 28 = 5 < 28	P28 = 107	137 − 107 = 30	263 − 107 = 156
11	33	35 − 33 = 2 < 33	P33 = 137	149 − 137 = 12	317 − 137 = 180
12	35	41 − 35 = 6 < 35	P35 = 149	179 − 149 = 30	349 − 149 = 200
13	41	43 − 41 = 2 < 41	P41 = 179	191 − 179 = 12	421 − 179 = 242
14	43	45 − 43 = 2 < 43	P43 = 191	197 − 191 = 6	443 − 191 = 252
15	45	49 − 45 = 4 < 45	P45 = 197	227 − 197 = 30	463 − 197 = 266
16	49	52 − 49 = 3 < 49	P49 = 227	239 − 227 = 12	521 − 227 = 294
17	52	57 − 52 = 5 < 52	P52 = 239	269 − 239 = 30	569 − 239 = 330
18	57	60 − 57 = 3 < 57	P57 = 269	281 − 269 = 12	619 − 269 = 350
19	60	64 − 60 = 4 < 60	P60 = 281	311 − 281 = 30	659 − 281 = 378
20	64	69 − 64 = 5 < 64	P64 = 311	347 − 311 = 36	719 − 311 = 408
21	69	81 − 69 = 12 < 69	P69 = 347	419 − 347 = 72	787 − 347 = 440
22	81	83 − 81 = 2 < 81	P81 = 419	431 − 419 = 12	953 − 419 = 534
23	83	89 − 83 = 6 < 83	P83 = 431	461 − 431 = 30	983 − 431 = 552
24	89	98 − 89 = 9 < 89	P89 = 461	521 − 461 = 60	1061 − 461 = 600
25	98	104 − 98 = 6 < 98	P98 = 521	569 − 521 = 48	1193 − 521 = 672
26	104	109 − 104 = 5 < 104	P104 = 569	599 − 569 = 30	1283 − 569 = 714
27	109	113 − 109 = 4 < 109	P109 = 599	617 − 599 = 18	1361 − 599 = 762
28	113	116 − 113 = 3 < 113	P113 = 617	641 − 617 = 24	1429 − 617 = 812
29	116	120 − 116 = 4 < 116	P116 = 641	659 − 641 = 18	1459 − 641 = 818
30	120	140 − 120 = 20 < 120	P120 = 659	809 − 659 = 150	1511 − 659 = 852
31	140	142 − 140 = 2 < 140	P140 = 809	821 − 809 = 12	1811 − 809 = 1002
32	142	144 − 142 = 2 < 142	P142 = 821	827 − 821 = 6	1861 − 821 = 1040
33	144	148 − 144 = 4 < 144	P144 = 827	857 − 827 = 30	1877 − 827 = 1050
34	148	152 − 148 = 4 < 148	P148 = 857	881 − 857 = 24	1949 − 857 = 1092
35	152	171 − 152 = 19 < 152	P152 = 881	1019 − 881 = 138	2003 − 881 = 1122
36	171	173 − 171 = 2 < 171	P171 = 1019	1031 − 1019 = 12	2297 − 1019 = 1278
37	173	176 − 173 = 3 < 173	P173 = 1031	1049 − 1031 = 18	2339 − 1031 = 1308
38	176	178 − 176 = 2 < 176	P176 = 1049	1061 − 1049 = 12	2377 − 1049 = 1328
39	178	182 − 178 = 4 < 178	P178 = 1061	1091 − 1061 = 30	2393 − 1061 = 1332
40	182	190 − 182 = 8 < 182	P182 = 1091	1151 − 1091 = 60	2459 − 1091 = 1368
41	190	201 − 190 = 11 < 190	P190 = 1151	1229 − 1151 = 78	2617 − 1151 = 1466
42	201	206 − 201 = 5 < 201	P201 = 1229	1277 − 1229 = 48	2753 − 1229 = 1524
43	206	209 − 206 = 3 < 206	P206 = 1277	1289 − 1277 = 12	2837 − 1277 = 1560
44	209	212 − 209 = 3 < 209	P209 = 1289	1301 − 1289 = 12	2887 − 1289 = 1598
45	212	215 − 212 = 3 < 212	P212 = 1301	1319 − 1301 = 18	2939 − 1301 = 1638
46	215	225 − 215 = 10 < 215	P215 = 1319	1427 − 1319 = 108	2999 − 1319 = 1680
47	225	230 − 225 = 5 < 225	P225 = 1427	1451 − 1427 = 24	3181 − 1427 = 1754
48	230	234 − 230 = 4 < 230	P230 = 1451	1481 − 1451 = 30	3257 − 1451 = 1806
49	234	236 − 234 = 2 < 234	P234 = 1481	1487 − 1481 = 6	3323 − 1481 = 1842
50	236	253 − 236 = 17 < 236	P236 = 1487	1607 − 1487 = 120	3347 − 1487 = 1860

Table 9 verifies C(i + 1) − C(i) < C(i) and P_C(i+1) − P_C(i) < P_2C(i) − P_C(i) for i ≤ 50. Because C(i + 1) − C(i) < C(i) and P_C(i+1) − P_C(i) < P_2C(i) − P_C(i) are two corollaries of Conjecture 4.5 and the conjecture has been verified up to n = 300,000 in Figure 3, the two corollaries have also been verified up to n = 300,000.

Note that there is no counterexample in all verifying results up to n = 300,000 at Step 3 of our model. Thus, if Conjecture 4.5 is proven then twin prime conjecture is true.

5. Further Discussion on Bertrand-Type Problem

5.1. Density of Second Order Primes

As we know, double prime, nontrivial prime and twin prime are second order primes. Correspondingly, {x} are natural numbers but {n} are called second order natural numbers. Therefore, we can contrast asymptotic average density of second order primes on x-axis and n-axis. Let D^prim(x) and D^prim(n) denote density of primes on x-axis and n-axis. Let D^doub(x) and D^doub(n) denote density of double primes on x-axis and n-axis. Let D^nont(x) and D^nont(n) denote density of nontrivial primes on x-axis and n-axis. Let D^twin(x) and D^twin(n) denote density of twin primes on x-axis and n-axis. Then Table 10 gives all forms of these densities.

Table 10. Density of primes and second order primes on x-axis and n-axis.

type of prime	x-axis	n-axis
prime	Dprim(x) ~ 1/log x	Dprim(n) = 1
double prime	Ddoub(x) ~ 1/(log x)2	Ddoub(n) ~ 1/log n
nontrivial prime	Dnont(x) ~ 1/(log x)2	Dnont(n) ~ 1/log n
twin prime	Dtwin(x) ~ 2C2/(log x)2	Dtwin(n) ~ 2C2/log n

Table 10 means Bertrand theorem suitable for primes on x-axis is not suitable for above three kinds of second order primes because their density is too low to support the theorem for second order primes on x-axis. However, Density of the three kinds of second order primes on n-axis is enough high to generalize Bertrand theorem into the prime index sequence as Bertrand-type theorem or conjecture. It is clear that P_2n > 2P_n for n > 1, which means P_2n − 2P_n > 0 for n > 1. By prime number theorem,

$P_{2n}-2P_n~2n\log \left(2n\right)-2n\log n=2n\log 2$ . (5.1)

In non-asymptotic case, we have the following approximation.

$P_{2n}-2P_n\approx 2n\log 2$ . (5.2)

Thus, we obtain

$P_{2n}-P_n\approx P_n+2n\log 2$ . (5.3)

Table 11 shows P_n + 2nlog2 is a good approximation for P_2n − P_n in non-asymptotic case. Because second order prime gap is bounded by P_2n − P_n, approximately, second order prime gap is bounded by P_n + 2nlog2. Thus, we have P_X(i+1) − P_X(i) < P_X(i) + 2X(i)log2 for a given second order prime P_X(i).

Let P_X(i) be a double prime P_A(i). Then we have the following approximate bound for double prime gap.

$P_A{}_{\left(i+1\right)}-P_A{}_{\left(i\right)}<P_A{}_{\left(i\right)}+2A\left(i\right)\log 2$ , (5.4)

$P_A{}_{\left(i+1\right)}<2P_A{}_{\left(i\right)}+2A\left(i\right)\log 2$ . (5.5)

Let P_X(i) be a nontrivial prime P_B(i). Then we have the following approximate bound for nontrivial prime gap.

$P_B{}_{\left(i+1\right)}-P_B{}_{\left(i\right)}<P_B{}_{\left(i\right)}+2B\left(i\right)\log 2$ , (5.6)

$P_B{}_{\left(i+1\right)}<2P_B{}_{\left(i\right)}+2B\left(i\right)\log 2$ . (5.7)

Let P_X(i) be a twin prime P_C(i). Then we have the following approximate bound for twin prime gap.

$P_C{}_{\left(i+1\right)}-P_C{}_{\left(i\right)}<P_C{}_{\left(i\right)}+2C\left(i\right)\log 2$ , (5.8)

$P_C{}_{\left(i+1\right)}<2P_C{}_{\left(i\right)}+2C\left(i\right)\log 2$ . (5.9)

Table 11. Comparison between accurate and approximate values for P_2n − P_n.

n	P2n − Pn	Pn + 2nlog2	relative error
2	4	5.772	0.3069
3	8	9.158	0.1264
5	18	17.93	0.0038
7	26	26.70	0.0262
10	42	42.86	0.0200
100	682	679	0.0043
1000	9470	9305	0.0174
10000	120008	118589	0.0118
100000	1450450	1438309	0.0083
1000000	16966980	16871863	0.0056
10000000	194163210	193284673	0.0045

Comparing with corollary of Bertrand theorem such that P_n+1 − P_n < P_n, approximations (5.4), (5.6) and (5.8) expanded the interval length so that such intervals are suitable for density of second order primes on natural number axis as Table 10 shows. Suppose P_A(i+1) − P_A(i) < P_A(i). Then we discover P_A(3) − P_A(2) = 11 − 5 = 6 > P_A(2) = 5. It means there is no double prime in (P_A(2), 2P_A(2)). The counterexample for i = 2 negated this hypothesis. However, using (5.4), we have P_A(3) − P_A(2) = 6 < P_A(2) + 6log2 = 9.158. Thus, the bound (5.4) is reasonable for controlling double prime gap on x-axis. Suppose P_C(i+1) − P_C(i) < P_C(i). Then we discover P_C(3) − P_C(2) = 11 − 5 = 6 > P_C(2) = 5. It means there is no twin prime in (P_C(2), 2P_C(2)). The counterexample for i = 2 negated this hypothesis. However, using (5.8), we have P_C(3) − P_C(2) = 6 < P _C(2) + 6log2 = 9.158. Thus, the bound (5.8) is reasonable for controlling twin prime gap on x-axis.

5.2. Ramanujan-Type Second Order Prime Root

Can one require that the second order natural number interval (n/2, n) contains not just at least one second order prime root but at least m second order prime roots? It leads to some results similar to Ramanujan primes.

Let π(n) denote the number of double prime roots among the first n prime roots. Then $R_m^{doub}$ is called the m-th Ramanujan-type double prime root if $R_m^{doub}$ satisfies

$\pi \left(n\right)-\pi \left(n/2\right)\ge m$ for all $n\ge R_m^{doub}$ . (5.10)

Since $R_1^{doub}=2$ , Bertrand-type theorem for double prime is the first Ramanujan-type double prime root case. By calculating π(n) − π(n/2) for n > 1, we have found the first five Ramanujan-type double prime roots 2, 11, 17, 29, 41 to correspond to m = 1, 2, 3, 4, 5. They are just the first five Ramanujan primes 2, 11, 17, 29, 41.

Let Ω(n) denote the number of nontrivial prime roots among the first n prime roots. Then $R_m^{nont}$ is called the m-th Ramanujan-type nontrivial prime root if $R_m^{nont}$ satisfies

$\Omega \left(n\right)-\Omega \left(n/2\right)\ge m$ for all $n\ge R_m^{nont}$ . (5.11)

Since $R_1^{nont}=2$ , Bertrand-type conjecture for nontrivial prime is the first Ramanujan-type nontrivial prime root case. By calculating Ω(n) − Ω(n/2) for n > 1, we have found the first five Ramanujan-type nontrivial prime roots 2, 3, 5, 7, 11 to correspond to m = 1, 2, 3, 4, 5. One can expect there are infinitely many Ramanujan-type nontrivial prime roots to show infinitude of Ramanujan-type nontrivial primes.

Let π₂(n) denote the number of twin prime roots among the first n prime roots. Then $R_m^{twin}$ is called the m-th Ramanujan-type twin prime root if $R_m^{twin}$ satisfies

${\pi }_2\left(n\right)-{\pi }_2\left(n/2\right)\ge m$ for all $n\ge R_m^{twin}$ . (5.12)

Since $R_1^{twin}=2$ , Bertrand-type conjecture for twin prime is the first Ramanujan-type twin prime root case. By calculating π₂(n) − π₂(n/2) for n > 1, we have found the first five Ramanujan-type twin prime roots 2, 7, 17, 28, 41 to correspond to m = 1, 2, 3, 4, 5. One can expect there are infinitely many Ramanujan-type twin prime roots to show infinitude of Ramanujan-type twin primes.

These results mean an existence problem of second order prime can be transformed into a problem of local density for second order prime because every second order prime root must link with a second order prime.

5.3. Asymptotic Form of Bertrand-Type System for Second Order Prime

We have known that, in non-asymptotic case, there are three corollaries which suggest P_X(i+1) is bounded by P_2X(i) and gap P_X(i+1) − P_X(i) is bounded by P_2X(i) − P_X(i) for a given second order prime P_X(i). However, in asymptotic case, we have

$\frac{P_{2n}}{P_n}~\frac{2n\log \left(2n\right)}{n\log n}=2+\frac{2\log 2}{\log n}$ .

Since $\frac{2\log 2}{\log n}~0$ , we obtain $\frac{P_{2n}}{P_n}~2$ . Thus, we get

$P_{2n}~2P_n$ . (5.13)

Result (5.13) means there are the following asymptotic forms.

For Bertrand-type theorem for double prime, we have

$P_A{}_{\left(i+1\right)}<P_{2A}{}_{\left(i\right)}~P_A{}_{\left(i+1\right)}<2P_A{}_{\left(i\right)}$ , (5.14)

$P_A{}_{\left(i+1\right)}-P_A{}_{\left(i\right)}<P_2{{}_A}_{\left(i\right)}-P_A{}_{\left(i\right)}~P_A{}_{\left(i+1\right)}-P_A{}_{\left(i\right)}<P_A{}_{\left(i\right)}$ . (5.15)

For Bertrand-type conjecture for nontrivial prime, we have

$P_B{}_{\left(i+1\right)}<P_{2B}{}_{\left(i\right)}~P_B{}_{\left(i+1\right)}<2P_B{}_{\left(i\right)}$ , (5.16)

$P_B{}_{\left(i+1\right)}-P_B{}_{\left(i\right)}<P_2{{}_B}_{\left(i\right)}-P_B{}_{\left(i\right)}~P_B{}_{(i+1)}-P_B{}_{\left(i\right)}<P_B{}_{\left(i\right)}$ . (5.17)

For Bertrand-type conjecture for twin prime, we have

$P_C{}_{\left(i+1\right)}<P_{2C}{}_{\left(i\right)}~P_C{}_{\left(i+1\right)}<2P_C{}_{\left(i\right)}$ , (5.18)

$P_C{}_{\left(i+1\right)}-P_C{}_{\left(i\right)}<P_2{{}_C}_{\left(i\right)}-P_C{}_{\left(i\right)}~P_C{}_{\left(i+1\right)}-P_C{}_{\left(i\right)}<P_C{}_{\left(i\right)}$ . (5.19)

These asymptotic forms mean the relative error between P_2X(i) and 2P_X(i) approaches 0 as i grows without bound, but we cannot use such asymptotic forms in non-asymptotic case as approximations.

6. Conclusion

This paper presents a standard model such that if counted number of a kind of second order primes is greater than counted number of double primes for n = 10⁹ then a Bertrand-type conjecture for the kind of second order primes can be established and the conjecture may imply the infinitude of the kind of second order primes. Based on the model, nontrivial primes and twin primes are considered. Goldbach conjecture has been transformed into an existence problem of nontrivial prime on n-axis by establishing Bertrand-type conjecture for nontrivial prime. The Bertrand-type conjecture can give proof of Goldbach conjecture. By establishing Bertrand-type conjecture for twin prime, twin prime conjecture has also become an existence problem of twin prime on n-axis. The Bertrand-type conjecture can also give proof of twin prime conjecture. In Bertrand-type system, it is a basic step that prime index n is defined as root of prime, which leads to the core significance: Bertrand-type form on n-axis remains linear control on second order prime root gap. The key result is: P_A(i+1) − P_A(i) is bounded by P_2A(i) − P_A(i), P_B(i+1) − P_B(i) is bounded by P_2B(i) − P_B(i) and P_C(i+1) − P_C(i) is bounded by P_2C(i) − P_C(i) on x-axis and these controls on second order prime gap are nonlinear, which is obviously different from linear control on prime gap as Bertrand theorem itself shows on natural number axis.

Acknowledgements

The author would like to acknowledge Rong Ao for his careful and helpful calculation and verification of the data.