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Institute for Basic Research, Palm Harbor, P.O. Box 3924, Beijing 100854, P.R. China

- *Corresponding Author:
- Chun-Xuan Jiang

Institute for Basic Research

Palm Harbor, P.O. Box 3924

Beijing 100854, P.R. China

**Tel:**+1-727-688 3992

**E-mail:**[email protected]

**Received Date:** March 10, 2017; **Accepted Date:** July 29, 2017; **Published Date:** July 31, 2017

**Citation: **Jiang CX (2017) Santilli’s Prime Chains: *P _{j}* + 1=

**Copyright:** © 2017 Jiang CX. This is an open-access article distributed under the
terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author and
source are credited.

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

Santilli’s prime chains: 1 j j P + = aP ± b , j=1,,k−1, (a,b)=1, 2|ab. If 1 1 n n n a − = Pλ Pλ , P1 Pn|b, we have J2(ω)→∞ as ω→∞. There exist infinitely many primes P1 such that P2,, Pk are primes for arbitrary length k. It is the Book proof. This is a generalized Euclid-Euler proof for the existence of infinitely many primes. Therefore Euclid- Euler-Jiang theorem in the distribution of primes is advanced. It is the Book theorem.

Generalized; Arithmetic; Prime; Euclid; Arbitrary; Integer

A new branch of number theory: Santilli’s additive isoprime theory is introduced. By using the arithmetic function Jn(ω) the following prime theorems have been proved.

1. There exist infinitely many twin primes.

2. The Goldbach’s theorem. Every even number greater than 4 is the sum of two odd primes.

3. There exist finitely many Mersenne primes, that is, primes of the
form 2* ^{P}*–1 where

4. There exist finitely many Fermat primes, that is, primes of the form .

5. There exist finitely many repunit primes whose digits (in base 10) are all ones.

6. There exist infinitely many primes of the forms: *x*^{2}+1, *x*^{4}+1, *x*^{8}+1, *x*^{16}+1, *x*^{32}+1, *x*^{64}+1.

7. There exist infinitely many primes of the forms: *x*^{2}+b, *x*^{3}+2, *x*^{5}+2, *x*^{7}+2

8. There exist infinitely many prime *m*-chains, *P _{j}*

9. There exist infinitely many triplets of consecutive integers, each
being the product of *k* distinct primes, (Here is an example: 1727913=3
× 11 × 52361, 1727914=2 × 17 × 50821, 1727915=5 × 7 × 49369.)

10. There exist infinitely many *k*-tuples of consecutive integers,
each being the product of m primes, where *k*>3, *m*>2.

11. Every integer m may be written in infinitely many ways in the form.

Where *k*=1,2,3,…,, *P*_{1} and *P*_{2} are primes.

12. There exist infinitely many Carmichael numbers, which are the product of three primes, four primes, and five primes.

13. There exist infinitely many prime chains in the arithmetic progressions.

14. In a table of prime numbers there exist infinitely many k-tuples
of primes, where *k*=2, 3, 4, …, 10^{5}.

15. Proof of Schinzel’s hypothesis.

16. Every large even number is representable in the form *P*_{1}+*P*_{2}…*P _{n}*.
It is the

17. Diophantine equation

, has infinitely many prime solutions.

18. There are infinitely many primes of the forms: *x*^{2} + *y ^{n}*,

19. There are infinitely many prime 5-tuples represented by P^{6}−42^{6}=(*P*−42)(*P*+42)(*P*^{2}+42*P*+1764)(*P*^{2}−42*P*+1764)

20. There are infinitely many prime k-tuples represented by *P ^{m}* ±

In this paper by using the arithmetic function *J*_{2}(ω) santilli’s
prime chains: *P _{j}*

**Theorem 1**

An increasing sequence of primes *P*_{1}, *P*_{2},…,*P _{k}* is called a Santilli’s
prime chain of the first kind of length

*P _{j}*

for *j*=1,…,*k*−1, (*a*,*b*)=1,2|*ab*.

We have the arithmetic function [6]

Where is called the primorials, *P _{i}* the last prime of the
primorials.

We now calculate *χ*(*P*). The smallest positive integer such that, *a ^{s}*=1(mod

*χ*(*P*)=*k* if *k*<*s*; *χ*(*P*)=*s* if *k*≥*s*; *χ*(*P*)=1 if *P*|*ab*

If *J*_{2}(ω)=0, there exist finitely many primes *P*_{1} such that *P*_{2},…, *P _{k}* are
primes for arbitrary length k. If

We have the best asymptotic formula of the number of primes *P*_{1}≤*N*,

Where is called the Euler function of the primorials.

The *P _{j+1}*=

**Theorem 2**

*P _{j+1}*=

We have the arithmetic function [6].

We now calculate *χ*(*P*). The smallest positive integer s such that,

2* ^{s}*≡1(mod

*χ*(*P*)=*k* if *k*<*s*; *χ*(*P*)=*s* if *k*≥*s*; *χ*(*P*)=1 if *P*|*b*.

Since *J*_{3}(ω)→ as ω→∞, there exist infinitely many primes *P*_{1} such
that P2,…,Pk are primes for arbitrary length *k*. This is the Book proof.

We have the best asymptotic formula of the number of primes *P*_{1} ≤ *N*.

The *P _{j+1}*=

**Example 1:***P _{j+1}*=

We have the arithmetic function,

Where *χ*(31)=−1, *χ*(*p*)=0 otherwise.

Since *J*_{2}(ω)→ as ω→∞, there exist infinitely many primes *P*_{1} such
that *P*_{2},…,*P*_{6} are primes.

We have the best asymptotic formula of the number of primes *P*_{1}≤*N*,

**Theorem 3**

*P _{j+1}*=3

We have the arithmetic function,

We now calculate *χ*(*P*). The smallest positive integer *s* such that,

3* ^{s}*≡1(mod

Since *J*_{2}(ω)→∞ as ω→∞, there exist infinitely many primes *P*_{1} such
that *P*_{2},…,*P _{k}* are primes for arbitrary length k.

We have the best asymptotic formula of the number of primes *P*_{1} ≤ *N*,

**Example 2:** *P _{j}*

We have the arithmetic function,

Since *J*_{2}()→∞ as ω→∞, there exist infinitely many primes *P*_{1} such
that *P*_{2},…,*P*_{6} are primes.

We have the best asymptotic formula of the number of primes *P*_{1}≤*N*.

**Theorem 4**

*P _{j}*

(1) 3|*b*, we have the arithmetic function,

We now calculate *χ*(*P*). The smallest positive integer *s* such that,

4* ^{s}*≡1(mod

*χ*(*P*)=*k* if *k*<*s*; *χ*(*P*)=*s* if *k*≥*s*; *χ*(*P*)=1 if *P*|*b*.

Since *J*_{2}(ω)→ as ω→∞, there exist infinitely many primes *P*_{1} such
that *P*_{2},…,*P _{k}* are primes for arbitrary length

We have the best asymptotic formula of the number of primes *P*_{1}≤*N*,

(2) *b*≠3*c* , *k*=3, we have *J*_{2}(3)=0.

(3) *b*3*c* , *k*=2 , we have *P*_{2}=4*P*_{1} ± *b*. Since *J*_{2}(ω)→∞ as ω→∞, there
exist infinitely many primes *P*_{1} such that *P*_{2} is a prime.

**Theorem 5**

*P _{j}*

We have the arithmetic function,

We now calculate *χ(P*). The smallest positive integer *s* such that

5* ^{s}*≡1(mod

*χ*(*P*)=*k* if *k*<*s*; *χ*(*P*)=s if *k*≥*s*; *χ*(5)=1; *χ*(*P*)=1 if *P*|*b*.

Since *J*_{2}(ω)→ as ω→∞, there exist infinitely many primes *P*_{1} such
that *P*_{2},…,*P _{k}* are primes for arbitrary length k.

We have the best asymptotic formula of the number of primes *P*_{1}≤*N*,

**Theorem 6**

*P _{j}*

(1) 5|*b*, we have the arithmetic function

We now calculate *χ*(*P*). The smallest positive integer *s* such that

6* ^{s}*≡1(mod

*χ*(*P*)=*k* if *k*<*s*; *χ*(*P*)=*s* if *k*≥*s*; *χ*(3)=1; *χ*(*P*)=1 if *P*|*b*.

Since *J*_{2}(ω)→ as ω→∞, there exist infinitely many primes *P*_{1} such
that *P*_{2},…,*P _{k}* are primes for arbitrary length

We have the best asymptotic formula of the number of primes *P*_{1}≤*N*,

(2) *b*≠5*c* , *k*=5, we have *J*_{2}(5)=0.

(3) *b*≠5*c*, *k*≤4 , we have *J*_{2}(ω)→∞ as ω→∞.

**Theorem 7**

*P _{j}*

(1) 6|b, we have *J*_{2}(ω)∞ as ω→∞, there exist infinitely many primes *P*_{1} such that *P*_{2},…,*P _{k}* are primes for arbitrary length

(2) 6|*b*, *k*=3, we have *J*_{2}(3)=0.

(3) 6|*b*, *k*=2 , we have *J*_{2}(ω)→∞ as ω→∞.

**Theorem 8**

*P _{j}*

(1) 7|*b*, we have *J*_{2}(ω)∞ as ω→∞, there exist infinitely many primes *P*_{1} such that *P*_{2},…,*P _{k}* are primes for arbitrary length

(2) *b*≠7*c*, *k*=7, we have *J*_{2}(7)=0.

(3) *b*7*c*, *k*≤6 , we have *J*_{2}(ω)→∞ as ω→∞.

**Theorem 9**

*P _{j}*

We have *J*_{2}(ω)→ as ω→∞, there exist infinitely many primes *P*_{1} such that *P*_{2},…,*P _{k}* are primes for arbitrary length k.

**Theorem 10**

*P _{j}*

(1) 3|b, we have *J*_{2}(ω)∞ as ω→∞, there exist infinitely many primes *P*_{1} such that *P*_{2},…,*P _{k}* are primes for arbitrary length

(2) *b*≠3*c*, *k*=3, we have *J*_{2}(7)=0.

(3) *b*3*c*, *k*=2, we have *J*_{2}(ω)→∞ as ω→∞.

**Theorem 11**

*P _{j}*

(1) 5|*b*, we have *J*_{2}(ω)∞ as ω→∞, there exist infinitely many primes *P*_{1} such that *P*_{2},…,*P _{k}* are primes for arbitrary length

(2) *b*≠5*c*, *k*=5, we have *J*_{2}(5)=0.

(3) *b*5*c*, *k*≤4, we have *J*_{2}(ω)→∞ as ω→∞.

**Theorem 12**

*P _{j}*

(1) 11|b, we have *J*_{2}(ω)→∞ as ω→∞, there exist infinitely many
primes *P*_{1} such that *P*_{2},…,*P _{k}* are primes for arbitrary length k.

(2) *b*≠11*c*, *k*=11, we have *J*_{2}(11)=0.

(3) *b*11*c*, *k*≤10, we have *J*_{2}(ω)→∞ as ω→∞.

**Theorem 13**

*P _{j}*

(1) 15|*b*, we have *J*_{2}(ω)→∞ as ω→∞, there exist infinitely many
primes *P*_{1} such that *P*_{2},…,*P _{k}* are primes for arbitrary length

(2) *b*≠3*c*, *k*=3, we have *J*_{2}(3)=0. *k*=2, we have *J*_{2}(ω)∞ as ω→∞.

(3) 3|*b*, *k*=5, we have *J*_{2}(5)=0. *k*=4, we have *J*_{2}(ω)→ as ω→∞.

**Theorem 14**

*P _{j}*

We have *J*_{2}(ω)→ as ω→∞, there exist infinitely many primes *P*_{1} such that *P*_{2},…,*P _{k}* are primes for arbitrary length

**Theorem 15**

*P _{j}*

We have *J*_{2}(ω)→ as ω→∞, there exist infinitely many primes *P*_{1} such that *P*_{2},…,*P _{k}* are primes for arbitrary length

**Theorem 16**

*P _{j}*

(1) 3|*b*, we have *J*_{2}(ω)→ as ω→∞, there exist infinitely many primes *P*_{1} such that *P*_{2},…,*P _{k}* are primes for arbitrary length

(2) 3|*b*, *k*=3, we have *J*_{2}(3)=0. *k*=2, we have *J*_{2}(ω)→ as ω→∞.

**Theorem 17**

(1) 6|*b*, we have *J*_{2}(ω)→ as ω→∞, there exist infinitely many primes *P*_{1} such that *P*_{2},…,*P _{k}* are primes for arbitrary length

(2) *b*≠6*c*, *k*=3, we have *J*_{2}(3)=0. *k*=2, we have *J*_{2}(ω)∞ as ω→∞.

**Theorem 18**

*b* is an odd number.

(1) 15|*b*, we have *J*_{2}(ω)∞ as ω→∞, there exist infinitely many primes *P*_{1} such that *P*_{2},…,*P _{k}* are primes for arbitrary length k.

(2) *b*≠3*c*, *k*=3, we have *J*_{2}(3)=0. *k*=2, we have *J*_{2}(ω)∞ as ω→∞.

(3) *b*3*c*, *k*=5, we have *J*_{2}(5)=0. *k*≤4, we have *J*_{2}(ω)→∞ as ω→∞.

**Theorem 19**

*b* is an odd number.

(1) 105|b, we have *J*_{2}(ω)∞ as ω→∞, there exist infinitely many
primes *P*_{1} such that *P*_{2},…,*P _{k}* are primes for arbitrary length

(2) 3|*b*, *k*=5, we have *J*_{2}(5)=0. *k*≤4, we have *J*_{2}(ω)→∞ as ω→∞.

(3) 3|*b*, *k*=3, we have *J*_{2}(3)=0. *K*=2, we have *J*_{2}(ω)→∞ as ω→∞.

**Theorem 20**

*b* is an odd
number.

(1) 1155|b, we have J2(ω)∞ as ω→∞, there exist infinitely many
primes *P*_{1} such that *P*_{2},…,*P _{k}* are primes for arbitrary length

(2) 3|*b*, *k*=5, we have *J*_{2}(5)=0. *k*≤4, we have *J*_{2}(ω)→∞ as ω→∞.

(3) *b*3*c*, *k*=3, we have *J*_{2}(3)=0. *K*=2, we have *J*_{2}(ω)→∞ as ω→∞.

**Theorem 21**

*b* is an odd number.

(1) 4123|b, we have *J*_{2}(ω)∞ as ω→∞, there exist infinitely many
primes *P*_{1} such that *P*_{2},…,*P _{k}* are primes for arbitrary length

(2) 7|*b*, *k*=19, we have *J*_{2}(19)=0. *k*≤18, we have *J*_{2}(ω)→∞ as ω→∞.

(3) *b*7*c*, *k*=7, we have *J*_{2}(7)=0. *K*≤6, we have *J*_{2}(ω)→∞ as ω→∞.

**Theorem 22**

*P _{j}*

If we have *J*_{2}(ω)∞ as ω→∞. There
exist infinitely many primes *P*_{1} such that *P*_{2},…,*P _{k}* are primes for
arbitrary length

Around 300BC by using the equation

(ω+1,ω)=1 as ω→∞,

Euclid proved that there are infinitely many primes.

In 1748 by using the equation

Euler proved that there are infinitely many primes.

*J*_{2}(ω)→∞ as ω→∞.

Jiang has proved that there exist infinitely many primes *P*_{1} such
that *P*_{2},…,*P _{k}* are primes [1-10]. It is a generalization of Euclid-Euler
theorem. Therefore Euclid-Euler-Jiang theorem in the distribution of
primes is advanced. It is the Book theorem.

From ref. [6] we have:

Therefore we have the prime number theorem.

Where π(*N*) denotes the number of primes ≤*N*.

From ref. [6] we have:

Therefore we have the prime k-tuples theorem:

Where *π _{k}*(

If the arithmetic constant , that is *J*_{2}(ω)0, there
exist infinitely many primes *P*_{1} such that *P*_{2},,*P _{k}* are primes.

- Jiang CX (1996) On the Yu-Goldbach prime theorem. Guangxi Sciences (in Chinese) 3: 9-12.
- Jiang CX (1998) Foundations of Santilli’s isonumber theory 1. Algebras, Groups and Geometries 15: 351-393. MR2000c: 11214.
- Jiang CX (1998) Foundations of Santilli’s isonumber theory 2. Algebras, Groups and Geometries 15: 509-544.
- Jiang CX (1999) Foundations of Santilli’s isonumber theory. In: Fundamental open problems in sciences at the end of the millennium, T. Gill, K. Liu and E. Trell (Eds)13 Hadronic Press, USA pp: 105-139.
- Jiang CX (2001) Proof of Schinzel’s hypothesis, Algebras Groups and Geometries 18: 411-420.
- Jiang CX (2002) Foundations of Santilli’s isonumber theory with applications to new cryptograms, Fermat’s theorem and Goldbach’s conjecture. International Academic Press. www. i-b-r.org
- Jiang CX, Disproofs of Riemann hypothesis. To appear.
- Jiang CX, Prime theorem in Santilli’s isonumber theory. To appear.
- Jiang CX, Prime theorem in Santilli’s isonumber theory (II). To appear.
- Jiang CX, Diophantine equation has infinitely many prime solution. To appear.

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