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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Prime Distribution in Pythagorean Triples (1)

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: 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.

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Mini-Review

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

Pythagorean triples

Equation (1)

In comprime integers must be of the form:

Equation (2)

Where x and y are coprime integers.

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

Equation (3)

Let xy =1 and 1 a = x + y = P1 , we have,

Equation (4)

Equation (5)

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

Equation (6)

There are infinitely many primes P1 such that P2 is a prime.

Proof: We have Jiang function [1]

Equation (7)

where Equation is the number of solutions of congruence

q2 +1 ≡ 0(mod P), q =1,…,P −1 . (8)

From (8) we have,

Equation (9)

Substituting (9) into (7) we have

Equation (10)

Since J2(ω)≠0, we prove that there are infinitely many prime P1, such that P2 is a prime.

We have the best asymptotic formula [1].

Equation (11)

whereEquation.

Theorem 2: Let x + y = P1 and xy = P1 − 2 , we have a = P1(P1 − 2) and,

Equation (12)

Equation (13)

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

Equation (14)

There are infinitely many primes P1 such that P2 is a prime.

Proof: We have Jiang function [1]

Equation (15)

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

Equation (16)

From (16) we have,

Equation (17)

Substituting (17) into (15) we have,

Equation (18)

Since J2(ω)≠0, we prove that there are infinitely many prime P1 such that P2 is a prime.

We have the best asymptotic formula [1]

Equation (19)

Theorem 3: Let xy =1 and a = x + y = P12 , we have,

Equation (20)

There are infinitely many primes P1 such that P2 is a prime.

Proof: We have Jiang function [1],

Equation (21)

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

Equation (22)

From (22) we have,

Equation (23)

Since J2(ω)≠0, we prove that there are infinitely many prime P1 such that P2 is a prime.

We have the best asymptotic formula [1]:

Equation (24)

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

References

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