Medical, Pharma, Engineering, Science, Technology and Business

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:**j[email protected]

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

**Citation: **Jiang CX (2017) Prime Distribution in Pythagorean Triples (1). J
Generalized Lie Theory Appl 11: 276. doi: 10.4172/1736-4337.1000276

**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

Using Jiang function we study the prime distribution in Pythagorean triples.

**Pythagorean triples**

(1)

In comprime integers must be of the form:

(2)

Where *x* and *y* are coprime integers.

**Theorem 1:** From eqn. (2) we have,

(3)

Let *x* − *y* =1 and 1 *a* = *x* + *y* = *P*_{1} , we have,

(4)

(5)

From eqns. (4) and (5) we have,

(6)

There are infinitely many primes *P*_{1} such that *P*_{2} is a prime.

**Proof:** We have Jiang function [1]

(7)

where is the number of solutions of congruence

*q*^{2} +1 ≡ 0(mod *P*), *q* =1,…,*P* −1 . (8)

From (8) we have,

(9)

Substituting (9) into (7) we have

(10)

Since *J*_{2}(ω)≠0, we prove that there are infinitely many prime *P*_{1}, such that *P*_{2} is a prime.

We have the best asymptotic formula [1].

(11)

where.

**Theorem 2:** Let *x* + *y* = *P*_{1} and *x* − *y* = *P*_{1} − 2 , we have *a* = *P*_{1}(*P*_{1} − 2) and,

(12)

(13)

From eqns. (12) and (13) we have,

(14)

There are infinitely many primes *P*_{1} such that *P*_{2} is a prime.

**Proof:** We have Jiang function [1]

(15)

Where *χ*(*P*) is the number of solutions of congruence

(16)

From (16) we have,

(17)

Substituting (17) into (15) we have,

(18)

Since *J*_{2}(*ω*)≠0, we prove that there are infinitely many prime *P*_{1} such that *P*_{2} is a prime.

We have the best asymptotic formula [1]

(19)

**Theorem 3:** Let *x* − *y* =1 and *a* = *x* + *y* = P_{1}^{2} , we have,

(20)

There are infinitely many primes *P*_{1} such that *P*_{2} is a prime.

**Proof:** We have Jiang function [1],

(21)

Where *χ* (*P*) is the number of solutions of congruence,

(22)

From (22) we have,

(23)

Since *J*_{2}(ω)≠0, we prove that there are infinitely many prime *P*_{1} such that *P*_{2} is a prime.

We have the best asymptotic formula [1]:

(24)

These results are in wide use in biological, physical and chemical fields.

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