On the Set of Primitive Triples of Natural Numbers Satisfying the Diophantine Equation of Pythagor

Bagram Sibgatullovich Kochkarev

doi:doi:10.11648/j.dmath.20200501.11

| Peer-Reviewed

On the Set of Primitive Triples of Natural Numbers Satisfying the Diophantine Equation of Pythagor

Bagram Sibgatullovich Kochkarev

Published in International Journal of Discrete Mathematics (Volume 5, Issue 1)

Received: 31 October 2019 Accepted: 25 December 2019 Published: 23 April 2020

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

The main task considered in the article is to find the condition primitive integer solutions of the Diophantine Pithagorean equation x²+y²=z² It is known that for this purpose it is enough to find primive solution of x, y such that x is even and y is odd. In this paper, in particular, we proved that the z of a primitive solution is a Prime number of the form 4k+1. It is prove in this paper that any right triangle with integer side lengths has a hypotenuse equal to a Prime of the form 4k+1and we show with the help of the descent axiom how to find primitive solutions of x and y in this case. We divide the search for primitive solutions (x, y, z) of right triangles into two cases: 1) the hypotenuse of such triangles is a Prime number of the form 4k+1 and 2) the hypotenuse of such triangles is a composite number. In section 3 we use formulas known to the ancient Hindus to find primitive solutions of Pithagorean equations in cases where m and n2+n² is an compaund number ending in 5. To find primes ending in 3 and 7, we refer the reader to our paper, which presents algorithms for constructing all primes and twin primes. The proposed paper also presents a generalization of Euclid's fundamental result on the infinity of the set of Primes, namely, it is shown that all twin primes are in residue classes (1, 3), (2, 4), (4, 1), and there are infinity many such twins.

Published in	International Journal of Discrete Mathematics (Volume 5, Issue 1)
DOI	10.11648/j.dmath.20200501.11
Page(s)	1-3
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Prime Numbers, Binary Problem, Axiom of Descent

References

[1]	Kochkarev B. S. K metodu spuska Ferma. Problems of Modern Science and Education, 2015, 11 (41), pp. 7-10 (in Russian).
[2]	Wiles A. Modular elliptic curves and Fermat's last theorem Annals of Mathematics, v, 141 Second series, 1995, pp. 445-551.
[3]	Kochkarev B. S. Problem of Recognition of Hamiltonian Graph. International Journal of Wirless Communication and Mobile Computing 2016, 4 (2), 52-55.
[4]	Kochkarev B. S. Vzaimootnosheniya mejdu slojnostnymi klassami P, NP i NPC. Problems of Modern Science and Education, 2015, 8 (35), 6-8. (in Russian).
[5]	Postnikov M. M. Vvedenie v teoriyu algebraicheskich chisel. Nauka, 1982 s. 240 (in Russian).
[6]	Buchshtab A. A. Teoriya chisel. Prosvetshenie. 1966 s. 384 (in Russian).
[7]	Singh S. Velikaya teorema Ferma. MTSHMO, 288 {in Russian).
[8]	Kochkarev B. S. Problema bliznetsov i drugie binarnye problemy. Problems of Modern Science and Education. 2015, 11 (41), s. 10-12. (in Russian).
[9]	Kochkarev B. S. About One Binary Problem in a Class of Algebraic Equation and Her Communication with the Great Hypothesis of Fermat. IJCMS, v. 2, Issue 10, pp. 457-459, October 2016.
[10]	Kochkarev B. S. Infinity Sequences of Prime of Form 4n-1 and 4n+1, International Journal of Humanities and Social Science Invention, v. 5, Issue 12, December 2016.
[11]	Kochkarev B. S. Algorithm of Search of Large Prime Numbers. International Journal of Discrete Mathematics, 2016, 1 (1) pp. 30-32.
[12]	Kochkarev B. S. An Binary Problem of Goldbach Euler and its Generalization. International Journal of Discrete Mathematics, 2918, 3 (2), pp. 32-35.
[13]	Kochkarev B. S. About Tenth Problem of D. Hilbert. AJER, Vol. 6, Issue, 12, 2017, pp. 241-242.
[14]	Kochkarev B. S. Axiom of Descent and Binary Mathematical Problem, AJER, 2018, Vol. 7, Issue, 2, pp. 117-118.
[15]	Kochkarev B, S. Zakonomernosti generatsii prostykh chisel i prostykh chisel bliznetsov. International Journal Chronos, 3 November, 2018, 52-53 (in Russian).

Cite This Article

Plain Text BibTeX RIS

APA Style

Bagram Sibgatullovich Kochkarev. (2020). On the Set of Primitive Triples of Natural Numbers Satisfying the Diophantine Equation of Pythagor. International Journal of Discrete Mathematics, 5(1), 1-3. https://doi.org/10.11648/j.dmath.20200501.11

Copy | Download

ACS Style

Bagram Sibgatullovich Kochkarev. On the Set of Primitive Triples of Natural Numbers Satisfying the Diophantine Equation of Pythagor. Int. J. Discrete Math. 2020, 5(1), 1-3. doi: 10.11648/j.dmath.20200501.11

Copy | Download

AMA Style

Bagram Sibgatullovich Kochkarev. On the Set of Primitive Triples of Natural Numbers Satisfying the Diophantine Equation of Pythagor. Int J Discrete Math. 2020;5(1):1-3. doi: 10.11648/j.dmath.20200501.11

Copy | Download

@article{10.11648/j.dmath.20200501.11,
  author = {Bagram Sibgatullovich Kochkarev},
  title = {On the Set of Primitive Triples of Natural Numbers Satisfying the Diophantine Equation of Pythagor},
  journal = {International Journal of Discrete Mathematics},
  volume = {5},
  number = {1},
  pages = {1-3},
  doi = {10.11648/j.dmath.20200501.11},
  url = {https://doi.org/10.11648/j.dmath.20200501.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20200501.11},
  abstract = {The main task considered in the article is to find the condition primitive integer solutions of the Diophantine Pithagorean equation x2+y2=z2 It is known that for this purpose it is enough to find primive solution of x, y such that x is even and y is odd. In this paper, in particular, we proved that the z of a primitive solution is a Prime number of the form 4k+1. It is prove in this paper that any right triangle with integer side lengths has a hypotenuse equal to a Prime of the form 4k+1and we show with the help of the descent axiom how to find primitive solutions of x and y in this case. We divide the search for primitive solutions (x, y, z) of right triangles into two cases: 1) the hypotenuse of such triangles is a Prime number of the form 4k+1 and 2) the hypotenuse of such triangles is a composite number. In section 3 we use formulas known to the ancient Hindus to find primitive solutions of Pithagorean equations in cases where m and n2+n2 is an compaund number ending in 5. To find primes ending in 3 and 7, we refer the reader to our paper, which presents algorithms for constructing all primes and twin primes. The proposed paper also presents a generalization of Euclid's fundamental result on the infinity of the set of Primes, namely, it is shown that all twin primes are in residue classes (1, 3), (2, 4), (4, 1), and there are infinity many such twins.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - On the Set of Primitive Triples of Natural Numbers Satisfying the Diophantine Equation of Pythagor
AU  - Bagram Sibgatullovich Kochkarev
Y1  - 2020/04/23
PY  - 2020
N1  - https://doi.org/10.11648/j.dmath.20200501.11
DO  - 10.11648/j.dmath.20200501.11
T2  - International Journal of Discrete Mathematics
JF  - International Journal of Discrete Mathematics
JO  - International Journal of Discrete Mathematics
SP  - 1
EP  - 3
PB  - Science Publishing Group
SN  - 2578-9252
UR  - https://doi.org/10.11648/j.dmath.20200501.11
AB  - The main task considered in the article is to find the condition primitive integer solutions of the Diophantine Pithagorean equation x2+y2=z2 It is known that for this purpose it is enough to find primive solution of x, y such that x is even and y is odd. In this paper, in particular, we proved that the z of a primitive solution is a Prime number of the form 4k+1. It is prove in this paper that any right triangle with integer side lengths has a hypotenuse equal to a Prime of the form 4k+1and we show with the help of the descent axiom how to find primitive solutions of x and y in this case. We divide the search for primitive solutions (x, y, z) of right triangles into two cases: 1) the hypotenuse of such triangles is a Prime number of the form 4k+1 and 2) the hypotenuse of such triangles is a composite number. In section 3 we use formulas known to the ancient Hindus to find primitive solutions of Pithagorean equations in cases where m and n2+n2 is an compaund number ending in 5. To find primes ending in 3 and 7, we refer the reader to our paper, which presents algorithms for constructing all primes and twin primes. The proposed paper also presents a generalization of Euclid's fundamental result on the infinity of the set of Primes, namely, it is shown that all twin primes are in residue classes (1, 3), (2, 4), (4, 1), and there are infinity many such twins.
VL  - 5
IS  - 1
ER  -

Copy | Download

Author Information

Bagram Sibgatullovich Kochkarev

Department of Mathematics and Mathematical Modeling, Institute of Mathematics and Mechanics Named After Nikolai Ivanovich Lobachevsky, Kazan (Volga Region) Federal University, Kazan, Russia

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Bagram Sibgatullovich Kochkarev. (2020). On the Set of Primitive Triples of Natural Numbers Satisfying the Diophantine Equation of Pythagor. International Journal of Discrete Mathematics, 5(1), 1-3. https://doi.org/10.11648/j.dmath.20200501.11

Copy | Download

ACS Style

Bagram Sibgatullovich Kochkarev. On the Set of Primitive Triples of Natural Numbers Satisfying the Diophantine Equation of Pythagor. Int. J. Discrete Math. 2020, 5(1), 1-3. doi: 10.11648/j.dmath.20200501.11

Copy | Download

AMA Style

Bagram Sibgatullovich Kochkarev. On the Set of Primitive Triples of Natural Numbers Satisfying the Diophantine Equation of Pythagor. Int J Discrete Math. 2020;5(1):1-3. doi: 10.11648/j.dmath.20200501.11

Copy | Download

@article{10.11648/j.dmath.20200501.11,
  author = {Bagram Sibgatullovich Kochkarev},
  title = {On the Set of Primitive Triples of Natural Numbers Satisfying the Diophantine Equation of Pythagor},
  journal = {International Journal of Discrete Mathematics},
  volume = {5},
  number = {1},
  pages = {1-3},
  doi = {10.11648/j.dmath.20200501.11},
  url = {https://doi.org/10.11648/j.dmath.20200501.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20200501.11},
  abstract = {The main task considered in the article is to find the condition primitive integer solutions of the Diophantine Pithagorean equation x2+y2=z2 It is known that for this purpose it is enough to find primive solution of x, y such that x is even and y is odd. In this paper, in particular, we proved that the z of a primitive solution is a Prime number of the form 4k+1. It is prove in this paper that any right triangle with integer side lengths has a hypotenuse equal to a Prime of the form 4k+1and we show with the help of the descent axiom how to find primitive solutions of x and y in this case. We divide the search for primitive solutions (x, y, z) of right triangles into two cases: 1) the hypotenuse of such triangles is a Prime number of the form 4k+1 and 2) the hypotenuse of such triangles is a composite number. In section 3 we use formulas known to the ancient Hindus to find primitive solutions of Pithagorean equations in cases where m and n2+n2 is an compaund number ending in 5. To find primes ending in 3 and 7, we refer the reader to our paper, which presents algorithms for constructing all primes and twin primes. The proposed paper also presents a generalization of Euclid's fundamental result on the infinity of the set of Primes, namely, it is shown that all twin primes are in residue classes (1, 3), (2, 4), (4, 1), and there are infinity many such twins.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - On the Set of Primitive Triples of Natural Numbers Satisfying the Diophantine Equation of Pythagor
AU  - Bagram Sibgatullovich Kochkarev
Y1  - 2020/04/23
PY  - 2020
N1  - https://doi.org/10.11648/j.dmath.20200501.11
DO  - 10.11648/j.dmath.20200501.11
T2  - International Journal of Discrete Mathematics
JF  - International Journal of Discrete Mathematics
JO  - International Journal of Discrete Mathematics
SP  - 1
EP  - 3
PB  - Science Publishing Group
SN  - 2578-9252
UR  - https://doi.org/10.11648/j.dmath.20200501.11
AB  - The main task considered in the article is to find the condition primitive integer solutions of the Diophantine Pithagorean equation x2+y2=z2 It is known that for this purpose it is enough to find primive solution of x, y such that x is even and y is odd. In this paper, in particular, we proved that the z of a primitive solution is a Prime number of the form 4k+1. It is prove in this paper that any right triangle with integer side lengths has a hypotenuse equal to a Prime of the form 4k+1and we show with the help of the descent axiom how to find primitive solutions of x and y in this case. We divide the search for primitive solutions (x, y, z) of right triangles into two cases: 1) the hypotenuse of such triangles is a Prime number of the form 4k+1 and 2) the hypotenuse of such triangles is a composite number. In section 3 we use formulas known to the ancient Hindus to find primitive solutions of Pithagorean equations in cases where m and n2+n2 is an compaund number ending in 5. To find primes ending in 3 and 7, we refer the reader to our paper, which presents algorithms for constructing all primes and twin primes. The proposed paper also presents a generalization of Euclid's fundamental result on the infinity of the set of Primes, namely, it is shown that all twin primes are in residue classes (1, 3), (2, 4), (4, 1), and there are infinity many such twins.
VL  - 5
IS  - 1
ER  -

Copy | Download