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

which yields the bound

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 xbe 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.

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.

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

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

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.

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 rootif 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 2.4Let P_A(i) denote the i-th double prime. Then g A ( i ) d o u b is called the i-th double primegap if g A ( i ) d o u b = 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 ( k ) ~ n ( log n ) k , corresponding to density: ~ 1/(logx)^k [30]. Thus, P n ( 2 ) = 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.

where

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

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.

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

If lim n → ∞ π ( n ) ∫ 2 n d t log t = 1 , then D d o u b ( n ) ~ 1 log n .

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

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.

Since it is assumed that

by (2.9) we have

The limit (2.10) means that

Since D d o u b ( n ) = π ( n ) n , we obtain

Hence the theorem holds.

Corollary 2.6.If lim n → ∞ π ( n ) ∫ 2 n d t log t = 1 , then there are infinitely many double primes.

Proof.By Theorem 2.5, if lim n → ∞ π ( n ) ∫ 2 n d t log t = 1

then we have

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 2.7When 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.

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_nand2n denote root of P_2n. Thenthere is at least one prime root n+k in(n,2n)for n> 1such that n+k is a double primeroot, correspondingly, there is at least one prime P_n+kin(P_n, P_2n)for n> 1such that P_n+kis 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

equivalently,

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.

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.

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.

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.

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

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

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

P₂, P₃, P₄, P₅, P₆, P₇, P₈, P₉, P₁₁, P₁₂P₁₃, 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 ofL_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.

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

where

is the logarithmic integral and it has an equivalent form

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

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.LetΩ(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 lim n → ∞ Ω ( n ) ∫ 2 n d t log t + n log n ( 1 log log n + 1 4 ( log log n ) 2 ) = 1 , then D n o n t ( n ) ~ 1 log n .

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

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.

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

Since it is assumed that

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

The limit (3.9) means that

Since D n o n t ( n ) = Ω ( n ) n , we obtain

Hence the proposition holds.

Corollary 3.7.If lim n → ∞ Ω ( n ) ∫ 2 n d t log t + n log n ( 1 log log n + 1 4 ( log log n ) 2 ) = 1 , then there are infinitely many nontrivial primes and Goldbach conjecture is true.

Proof.By Proposition 3.6, if lim n → ∞ Ω ( n ) ∫ 2 n d t log t + n log n ( 1 log log n + 1 4 ( log log n ) 2 ) = 1

then we have

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.

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.

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_nand2n denote root of P_2n. Then there is at least one prime root n+k in(n,2n)for n> 1such that n+kis a nontrivialprime root, correspondingly, there is at leastoneprimeP_n+kin(P_n, P_2n) for n> 1 such that P_n+kisanontrivial 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

equivalently,

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 +kin (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)andP_2B(i) − P_B(i) > P_B(i) for i ≥ 1.

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).

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.

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.