alexa Santilliand#8217;s Prime Chains: <em>P<sub>j</sub></em> + 1=<em>aP<sub>j</sub></em> and#177; <em>b</em> | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Santilli’s Prime Chains: Pj + 1=aPj ± b

Chun-Xuan Jiang*

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: Pj + 1=aPj ± b. J Generalized Lie Theory Appl 11: 280. doi: 10.4172/1736-4337.1000280

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.

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Abstract

Santilli’s prime chains: Pj + 1=aPj ± b , j=1,⋯,k−1, (a,b)=1, 2|ab. If a −1 = P1λn Pnλn , P1Pn|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.

Keywords

Generalized; Arithmetic; Prime; Euclid; Arbitrary; Integer

Introduction

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.

It is the Book proof. [1-10].

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 2P–1 where P is prime.

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

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

6. There exist infinitely many primes of the forms: x2+1, x4+1, x8+1, x16+1, x32+1, x64+1.

7. There exist infinitely many primes of the forms: x2+b, x3+2, x5+2, x7+2

8. There exist infinitely many prime m-chains, Pj+1=mPj±(m−1), m=2,3,…, including Cunningham chains.

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.

Equation

Where k=1,2,3,…,, P1 and P2 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, …, 105.

15. Proof of Schinzel’s hypothesis.

16. Every large even number is representable in the form P1+P2Pn. It is the n primes theorem which has no almost-primes.

17. Diophantine equation

Equation, has infinitely many prime solutions.

18. There are infinitely many primes of the forms: x2 + yn, n ≥2 and Equation .

19. There are infinitely many prime 5-tuples represented by P6−426=(P−42)(P+42)(P2+42P+1764)(P2−42P+1764)

20. There are infinitely many prime k-tuples represented by Pm ± Am.

In this paper by using the arithmetic function J2(ω) santilli’s prime chains: Pj+1=aPj ± b are studied. It is a generalization of santillis isoprime m-chains: Pj+1=mPj ± (m−1) [6].

Santilli’s Prime Chains: Pj+1=aPj ± b

Theorem 1

An increasing sequence of primes P1, P2,…,Pk is called a Santilli’s prime chain of the first kind of length k if

Pj+1=aPj+b

for j=1,…,k−1, (a,b)=1,2|ab.

We have the arithmetic function [6]

Equation

Where Equation is called the primorials, Pi the last prime of the primorials.

We now calculate χ(P). The smallest positive integer such that, as=1(mod P), (a, b)=1.

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

If J2(ω)=0, there exist finitely many primes P1 such that P2,…, Pk are primes for arbitrary length k. If 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 generalization of the Euclid-Euler proof for the existence of infinitely many primes.

We have the best asymptotic formula of the number of primes P1N,

Equation

Where Equation is called the Euler function of the primorials.

The Pj+1=aPjb is called a Santilli’s prime chain of the second kind of length k. Both Pj+1=aPj ± b have the same arithmetic function J2(ω). If a=m and b=m−1, it is Santilli’s isoprime m-chains [6].

Theorem 2

Pj+1=aPj ± b, j=1,…,k−1 , b is an odd number.

We have the arithmetic function [6].

Equation

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

2s≡1(mod P),

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

Since J3(ω)→ as ω→∞, there exist infinitely many primes P1 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 P1N.

Equation

The Pj+1=2Pj ± 1 are Cunningham prime chains [6].

Example 1:Pj+1=2Pj + 7, j=1,2,3,4,5.

We have the arithmetic function,

Equation

Where χ(31)=−1, χ(p)=0 otherwise.

Since J2(ω)→ as ω→∞, there exist infinitely many primes P1 such that P2,…,P6 are primes.

We have the best asymptotic formula of the number of primes P1N,

Equation

Theorem 3

Pj+1=3Pj ± b, j=1,…,k−1, (3,b)=1, 2|b.

We have the arithmetic function,

Equation

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

3s≡1(mod P),

Since J2(ω)→∞ as ω→∞, there exist infinitely many primes P1 such that P2,…,Pk are primes for arbitrary length k.

We have the best asymptotic formula of the number of primes P1N,

Equation

Example 2: Pj+1=3Pj±4, j=1, 2, 3, 4, 5.

We have the arithmetic function,

Equation

Since J2()→∞ as ω→∞, there exist infinitely many primes P1 such that P2,…,P6 are primes.

We have the best asymptotic formula of the number of primes P1N.

Equation

Theorem 4

Pj+1=4Pj±b, j=1,…,k−1, b is an odd number.

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

Equation

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

4s≡1(mod P).

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

Since J2(ω)→ as ω→∞, there exist infinitely many primes P1 such that P2,…,Pk are primes for arbitrary length k.

We have the best asymptotic formula of the number of primes P1N,

Equation

(2) b≠3c , k=3, we have J2(3)=0.

(3) b3c , k=2 , we have P2=4P1 ± b. Since J2(ω)→∞ as ω→∞, there exist infinitely many primes P1 such that P2 is a prime.

Theorem 5

Pj+1=5Pj±b, j=1,…,k−1,(5,b)=1, 2|b is an odd number.

We have the arithmetic function,

Equation

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

5s≡1(mod P).

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

Since J2(ω)→ as ω→∞, there exist infinitely many primes P1 such that P2,…,Pk are primes for arbitrary length k.

We have the best asymptotic formula of the number of primes P1N,

Equation

Theorem 6

Pj+1=6Pj±b, j=1,…,k−1,(3,b)=1, b is an odd number.

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

Equation

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

6s≡1(mod P).

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

Since J2(ω)→ as ω→∞, there exist infinitely many primes P1 such that P2,…,Pk are primes for arbitrary length k.

We have the best asymptotic formula of the number of primes P1N,

Equation

(2) b≠5c , k=5, we have J2(5)=0.

(3) b≠5c, k≤4 , we have J2(ω)→∞ as ω→∞.

Theorem 7

Pj+1=7Pj±b, j=1,…,k−1 , (7,b)=1, 2|b.

(1) 6|b, we have J2(ω)∞ as ω→∞, there exist infinitely many primes P1 such that P2,…,Pk are primes for arbitrary length k.

(2) 6|b, k=3, we have J2(3)=0.

(3) 6|b, k=2 , we have J2(ω)→∞ as ω→∞.

Theorem 8

Pj+1=8Pj±b, j=1,…,k−1 , b is an odd number.

(1) 7|b, we have J2(ω)∞ as ω→∞, there exist infinitely many primes P1 such that P2,…,Pk are primes for arbitrary length k.

(2) b≠7c, k=7, we have J2(7)=0.

(3) b7c, k≤6 , we have J2(ω)→∞ as ω→∞.

Theorem 9

Pj+1=9Pj±b, j=1,…,k−1 , (3,b)=1,2|b.

We have J2(ω)→ as ω→∞, there exist infinitely many primes P1 such that P2,…,Pk are primes for arbitrary length k.

Theorem 10

Pj+1=10Pj±b, j=1,…,k−1 , b is an odd number. (5,b)=1.

(1) 3|b, we have J2(ω)∞ as ω→∞, there exist infinitely many primes P1 such that P2,…,Pk are primes for arbitrary length k.

(2) b≠3c, k=3, we have J2(7)=0.

(3) b3c, k=2, we have J2(ω)→∞ as ω→∞.

Theorem 11

Pj+1=11Pj±b, j=1,…,k−1 , 2|b, (11,b)=1.

(1) 5|b, we have J2(ω)∞ as ω→∞, there exist infinitely many primes P1 such that P2,…,Pk are primes for arbitrary length k.

(2) b≠5c, k=5, we have J2(5)=0.

(3) b5c, k≤4, we have J2(ω)→∞ as ω→∞.

Theorem 12

Pj+1=12Pj±b, j=1,…,k−1, (3,b)=1, b is an odd number.

(1) 11|b, we have J2(ω)→∞ as ω→∞, there exist infinitely many primes P1 such that P2,…,Pk are primes for arbitrary length k.

(2) b≠11c, k=11, we have J2(11)=0.

(3) b11c, k≤10, we have J2(ω)→∞ as ω→∞.

Theorem 13

Pj+1=16Pj±b, j=1,…,k−1, b is an odd number.

(1) 15|b, we have J2(ω)→∞ as ω→∞, there exist infinitely many primes P1 such that P2,…,Pk are primes for arbitrary length k.

(2) b≠3c, k=3, we have J2(3)=0. k=2, we have J2(ω)∞ as ω→∞.

(3) 3|b, k=5, we have J2(5)=0. k=4, we have J2(ω)→ as ω→∞.

Theorem 14

Pj+1=17Pj±b, j=1,…,k−1, 2|b, (17, b)=1 is an odd number.

We have J2(ω)→ as ω→∞, there exist infinitely many primes P1 such that P2,…,Pk are primes for arbitrary length k.

Theorem 15

Pj+1=(2λ+1) Pj±b, j=1,…, k−1, 2|b, ((2λ+1), b)=1.

We have J2(ω)→ as ω→∞, there exist infinitely many primes P1 such that P2,…,Pk are primes for arbitrary length k.

Theorem 16

Pj+1=(3λ+1) Pj±b, j=1,…, k−1, 2|b, ((3n+1), b)=1 b is an odd number..

(1) 3|b, we have J2(ω)→ as ω→∞, there exist infinitely many primes P1 such that P2,…,Pk are primes for arbitrary length k.

(2) 3|b, k=3, we have J2(3)=0. k=2, we have J2(ω)→ as ω→∞.

Theorem 17

Equation

(1) 6|b, we have J2(ω)→ as ω→∞, there exist infinitely many primes P1 such that P2,…,Pk are primes for arbitrary length k.

(2) b≠6c, k=3, we have J2(3)=0. k=2, we have J2(ω)∞ as ω→∞.

Theorem 18

Equation b is an odd number.

Equation

(1) 15|b, we have J2(ω)∞ as ω→∞, there exist infinitely many primes P1 such that P2,…,Pk are primes for arbitrary length k.

(2) b≠3c, k=3, we have J2(3)=0. k=2, we have J2(ω)∞ as ω→∞.

(3) b3c, k=5, we have J2(5)=0. k≤4, we have J2(ω)→∞ as ω→∞.

Theorem 19

Equation b is an odd number.

Equation

(1) 105|b, we have J2(ω)∞ as ω→∞, there exist infinitely many primes P1 such that P2,…,Pk are primes for arbitrary length k.

(2) 3|b, k=5, we have J2(5)=0. k≤4, we have J2(ω)→∞ as ω→∞.

(3) 3|b, k=3, we have J2(3)=0. K=2, we have J2(ω)→∞ as ω→∞.

Theorem 20

Equation b is an odd number.

Equation

(1) 1155|b, we have J2(ω)∞ as ω→∞, there exist infinitely many primes P1 such that P2,…,Pk are primes for arbitrary length k.

(2) 3|b, k=5, we have J2(5)=0. k≤4, we have J2(ω)→∞ as ω→∞.

(3) b3c, k=3, we have J2(3)=0. K=2, we have J2(ω)→∞ as ω→∞.

Theorem 21

Equation b is an odd number.

Equation

(1) 4123|b, we have J2(ω)∞ as ω→∞, there exist infinitely many primes P1 such that P2,…,Pk are primes for arbitrary length k.

(2) 7|b, k=19, we have J2(19)=0. k≤18, we have J2(ω)→∞ as ω→∞.

(3) b7c, k=7, we have J2(7)=0. K≤6, we have J2(ω)→∞ as ω→∞.

Theorem 22

Pj+1=aPj ± b, j=1,…, k−1, (a, b)=1,2|ab.

If Equation we have J2(ω)∞ as ω→∞. There exist infinitely many primes P1 such that P2,…,Pk are primes for arbitrary length k[6].

Euclid-Euler-Jiang Theorem

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

Equation

Euler proved that there are infinitely many primes.

By using the equation [1-10].

J2(ω)→∞ as ω→∞.

Jiang has proved that there exist infinitely many primes P1 such that P2,…,Pk 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:

Equation

Therefore we have the prime number theorem.

Equation

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

From ref. [6] we have:

Equation

Therefore we have the prime k-tuples theorem:

Equation

Where πk(N,2) denotes the number of primes P1N.

If the arithmetic constant Equation, that is J2(ω)0, there exist infinitely many primes P1 such that P2,,Pk are primes. πk(N,2) have the same form Equation but differ in Ck.

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