RP 166 Solving Some Special Standard Cubic Congruence of Composite Modulus modulo a Multiple of an...
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International Journal of Tre Volume 5 Issue 4, May-June @ IJTSRD | Unique Paper ID – IJTSRD4 RP-166: Solving Som Composite Modulu Head, Department of Mathematics, Jag ABSTRACT Here in this paper, ten special type of composite modulus are studied for their s the cubic congruence under considerat solution can be obtained orally as the solu extra effort is necessary to find the solutio KEYWORDS: Cubic Congruence, Composite INTRODUCTION Some standard cubic congruence of s considered for study and are formulated All the considered cubic congruence solutions. Those solutions are present in the congru is the list of those cubic Congruence statement. PROBLEM-STATEMENT “To find formula for solutions of the congr ≡ 2, ≡ 3, ≡ 2 3, ≡ 4, ≡ 3 4, ≡ 6, ≡ 2 6, ≡ 3 6, ≡ 4 6, ≡ 5 6, . LITERATURE REVIEW The standard cubic congruence found n literature of mathematics as it is not stu part of syllabus in the university cour end in Scientific Research and Dev 2021 Available Online: www.ijtsrd.com e-I 42321 | Volume – 5 | Issue – 4 | May-June 2 me Special Standard Cubic C us modulo a Multiple of an Prof B M Roy gat Arts, Commerce & I H P Science College, Gor f standard cubic congruence of solutions. It is found that each of tion has a single solution. The ution is given in the problems. No on. Modulus, Unique Solution How to cit "RP-166: S Cubic Co Modulus m Prime" Pub Internation Journal of Scientific and Dev (ijtsrd), ISS 6470, Vo Issue-4, Ju pp.551-553 www.ijtsrd Copyright Internation Scientific Journal. Th distributed the terms Creative Attribution (http://creat special type are d their solutions. e have unique uence itself. Here in the problem ruence: . " no place in the udied; it is not a rse. Only linear congruence of degree on congruence of prime an remained in the part of stud author’s papers are seen [4], ANALYSIS & RESULTS Consider the congruence: odd prime. It is easily seen that: 0 8 as . Hence ≡ 2 is a so Consider the congruence: odd prime. It is easily seen that: 0 3 as . Hence ≡ 3 is a so Consider the congruence: odd prime. It is easily seen that: 2 . 3 ≡ 0 3 as Hence ≡ 2 3 is a s Consider the congruence: odd prime. It is easily seen that: 1 . 4 ≡ 0 4 as velopment (IJTSRD) ISSN: 2456 – 6470 2021 Page 551 Congruence of n Odd Prime regaon, Maharashtra, India te this paper: Prof B M Roy Solving Some Special Standard ongruence of Composite modulo a Multiple of an Odd blished in nal Trend in Research velopment SN: 2456- olume-5 | une 2021, 3, URL: d.com/papers/ijtsrd42321.pdf © 2021 by author (s) and nal Journal of Trend in Research and Development his is an Open Access article d under s of the Commons n License (CC BY 4.0) tivecommons.org/licenses/by/4.0) ne and standard quadratic nd composite modulus are dy [1], [2], [3].Also some of the [5], [6]. ≡ 2. Here p is an ! 1 . 2 ≡ olution of the congruence. ≡ 3. Here p is an 1" # 1 . 3 ≡ olution of the congruence. ≡ 2 3. Here p is an 2 22 12" # 1 . solution of the congruence. ≡ 4. Here p is an ! 1 1 # . IJTSRD42321
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Here in this paper, ten special type of standard cubic congruence of composite modulus are studied for their solutions. It is found that each of the cubic congruence under consideration has a single solution. The solution can be obtained orally as the solution is given in the problems. No extra effort is necessary to find the solution. Prof B M Roy "RP-166: Solving Some Special Standard Cubic Congruence of Composite Modulus modulo a Multiple of an Odd Prime" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-4 , June 2021, URL: https://www.ijtsrd.compapers/ijtsrd42321.pdf Paper URL: https://www.ijtsrd.commathemetics/applied-mathamatics/42321/rp166-solving-some-special-standard-cubic-congruence-of-composite-modulus-modulo-a-multiple-of-an-odd-prime/prof-b-m-roy
Transcript of RP 166 Solving Some Special Standard Cubic Congruence of Composite Modulus modulo a Multiple of an...
International Journal of Trend in Scientific Research and DevelopmentVolume 5 Issue 4, May-June
@ IJTSRD | Unique Paper ID – IJTSRD42321
RP-166: Solving Some Special Standard Cubic Congruence
Composite Modulus
Head, Department of Mathematics, Jagat Arts, Commerce &
ABSTRACT
Here in this paper, ten special type of standard cubic congruence of
composite modulus are studied for their solutions. It is found that each of
the cubic congruence under consideration has a single solution. The
solution can be obtained orally as the solu
extra effort is necessary to find the solution.
KEYWORDS: Cubic Congruence, Composite Modulus, Unique Solution
INTRODUCTION
Some standard cubic congruence of special type are
considered for study and are formulated their solutions.
All the considered cubic congruence have unique
solutions.
Those solutions are present in the congruence itself. Here
is the list of those cubic Congruence in the problem
statement.
PROBLEM-STATEMENT
“To find formula for solutions of the congruence:
�� ≡ ����2��,
�� ≡ ����3��,
�� ≡ 2����3��,
�� ≡ ����4��,
�� ≡ 3����4��,
�� ≡ ����6��,
�� ≡ 2����6��,
�� ≡ 3����6��,
�� ≡ 4����6��,
�� ≡ 5����6��,��������������.
LITERATURE REVIEW
The standard cubic congruence found no place in the
literature of mathematics as it is not studied; it is not a
part of syllabus in the university course. Only linear
Trend in Scientific Research and Development 2021 Available Online: www.ijtsrd.com e-ISSN: 2456
42321 | Volume – 5 | Issue – 4 | May-June 202
166: Solving Some Special Standard Cubic Congruence
Composite Modulus modulo a Multiple of an Odd Prime
Prof B M Roy
Jagat Arts, Commerce & I H P Science College, Goregaon
Here in this paper, ten special type of standard cubic congruence of
composite modulus are studied for their solutions. It is found that each of
the cubic congruence under consideration has a single solution. The
solution can be obtained orally as the solution is given in the problems. No
extra effort is necessary to find the solution.
Cubic Congruence, Composite Modulus, Unique Solution
How to cite this paper
"RP-166: Solving Some Special Standard
Cubic Congruence of Composite
Modulus modulo a Multiple of an Odd
Prime" Published in
International
Journal of Trend in
Scientific Research
and Development
(ijtsrd), ISSN: 2456
6470, Volume
Issue-4, June 2021,
pp.551-553,
www.ijtsrd.com/papers/ijtsrd42321.pdf
Copyright © 20
International Journal
Scientific Research and Development
Journal. This is an Open Access article
distributed un
the terms
Creative Commons
Attribution License(http://creativecommons.org/licenses/by/4.0
Some standard cubic congruence of special type are
considered for study and are formulated their solutions.
All the considered cubic congruence have unique
Those solutions are present in the congruence itself. Here
gruence in the problem
“To find formula for solutions of the congruence:
. "
The standard cubic congruence found no place in the
literature of mathematics as it is not studied; it is not a
part of syllabus in the university course. Only linear
congruence of degree one and standard quadratic
congruence of prime and composite modul
remained in the part of study [1], [2], [3].Also some of the
author’s papers are seen [4], [5], [6].
ANALYSIS & RESULTS
Consider the congruence: �
odd prime.
It is easily seen that:
0���8�� as ����.
Hence � ≡ ����2�� is a solution of the congruence.
Consider the congruence: �
odd prime.
It is easily seen that: �� � �
0���3�� as ����.
Hence � ≡ ����3�� is a solution of the congruence.
Consider the congruence: ��
odd prime.
It is easily seen that: �2��� �
�. 3� ≡ 0���3�� as ����
Hence � ≡ 2����3�� is a solution of the congruence.
Consider the congruence: �
odd prime.
It is easily seen that: �� � �
1� �. 4� ≡ 0���4�� as �
Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 – 6470
2021 Page 551
166: Solving Some Special Standard Cubic Congruence of
n Odd Prime
I H P Science College, Goregaon, Maharashtra, India
How to cite this paper: Prof B M Roy
Solving Some Special Standard
Cubic Congruence of Composite
Modulus modulo a Multiple of an Odd
Prime" Published in
International
Journal of Trend in
Scientific Research
and Development
(ijtsrd), ISSN: 2456-
6470, Volume-5 |
4, June 2021,
553, URL:
www.ijtsrd.com/papers/ijtsrd42321.pdf
Copyright © 2021 by author (s) and
International Journal of Trend in
Scientific Research and Development
Journal. This is an Open Access article
distributed under
the terms of the
Creative Commons
Attribution License (CC BY 4.0) http://creativecommons.org/licenses/by/4.0)
congruence of degree one and standard quadratic
congruence of prime and composite modulus are
remained in the part of study [1], [2], [3].Also some of the
author’s papers are seen [4], [5], [6].
� ≡ ����2��. Here p is an
�� � � ���! � 1� �. 2� ≡
is a solution of the congruence.
� ≡ ����3��. Here p is an
� ��� � 1��" # 1� �. 3� ≡
is a solution of the congruence.
� ≡ 2����3��. Here p is an
� 2� 2��2� � 1��2" # 1�
�.
is a solution of the congruence.
� ≡ ����4��. Here p is an
� ���! � 1� ��� � 1��� #
����.
IJTSRD42321
International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
@ IJTSRD | Unique Paper ID – IJTSRD42321 | Volume – 5 | Issue – 4 | May-June 2021 Page 552
Hence � ≡ ����4�� is a solution of the congruence.
Consider the congruence: �� ≡ 3����4��. Here p is an
odd prime.
It is easily seen that: �3��� � 3� 3��9�! � 1�
3�[4�! # 4�! # ��! � 1�]
3�. 4� ≡ 0���4��.
Hence � ≡ 3����4�� is a solution of the congruence.
Consider the congruence: �� ≡ ����8��. Here p is an
odd prime.
It is easily seen that: �� � � ���! � 1� �. 8� ≡
0���8�� as �! ≡ 1���8�.
Hence � ≡ ����8�� is a solution of the congruence
Consider the congruence: �� ≡ 3����8��. Here p is an
odd prime.
It is easily seen that: �3��� � 3� 3��9�! � 1�
3�[8�! # ��! � 1�]
3�. 8� ≡ 0���8��.
Hence � ≡ ����8�� is a solution of the congruence.
Consider the congruence:�� ≡ 5����8��. Here p is an
odd prime.
It is easily seen that: �5��� � 5� 5��25�! � 1�
5�[24�! # ��! � 1�]
5�. 8� ≡ 0���8��.
Hence � ≡ 5����8�� is a solution of the congruence.
Consider the congruence: �� ≡ 7����8��. Here p is an
odd prime.
It is easily seen that: �7��� � 7� 7��49�! � 1�
7�[48�! # ��! � 1�]
7�. 8� ≡ 0���8��.
Hence � ≡ 7����8�� is a solution of the congruence.
ILLUSTRATIONS
Example-1:Consider the congruence �� ≡ 7���14�.
It can be written as �� ≡ 7���2.7�.
It is of the type �� ≡ ����2��(��ℎ� 7.
It has single solution � ≡ ����2��
≡ 7���2.7�
≡ 7���14�.
Consider the congruence �� ≡ 7���21�.
It can be written as �� ≡ 7���3.7�.
It is of the type �� ≡ ����3��(��ℎ� 7.
It has single solution � ≡ ����3��
≡ 7���3.7�
≡ 7���21�.
Consider the congruence �� ≡ 14���21�.
It can be written as �� ≡ 2.7���3.7�.
It is of the type �� ≡ 2����3��(��ℎ� 7.
It has single solution � ≡ 2����3��
≡ 2.7���3.7�
≡ 14���21�.
Example-1: Consider the congruence �� ≡ 7���56�.
It can be written as �� ≡ 7���8.7�.
It is of the type �� ≡ ����8��(��ℎ� 7.
It has single solution � ≡ ����8��
≡ 7���8.7�
≡ 7���56�.
Example-2: Consider the congruence �� ≡ 21���56�.
It can be written as �� ≡ 3.7���8.7�.
It is of the type �� ≡ 3����8��(��ℎ� 7.
It has single solution � ≡ 3����8��
≡ 3.7���8.7�
≡ 21���56�.
Example-3: Consider the congruence �� ≡ 35���56�.
It can be written as �� ≡ 5.7���8.7�.
It is of the type �� ≡ 5����8��(��ℎ� 7.
It has single solution � ≡ 5����8��
≡ 5.7���8.7�
≡ 35���56�.
Example-4: Consider the congruence �� ≡ 49���56�.
It can be written as �� ≡ 7.7���8.7�.
It is of the type �� ≡ 7����8��(��ℎ� 7.
It has single solution � ≡ 7����8��
≡ 7.7���8.7�
≡ 49���56�.
CONCLUSION
It can be concluded from this discussion that the standard
cubic congruence considered, each has a single solutions.
It is found that the congruence �� ≡ ����2��, p an odd
prime has a unique solution
� ≡ ����2��.
The congruence �� ≡ ����3��, p an odd prime has a
unique solution
� ≡ ����3��.
The congruence �� ≡ 2����3��, p an odd prime has a
unique solution
� ≡ 2����3��.
The congruence �� ≡ ����4��, p an odd prime has a
unique solution
� ≡ ����4��.
The congruence �� ≡ 3����4��, p an odd prime has a
unique solution
� ≡ 3����4��.
The congruence �� ≡ ����8��, p an odd prime has a
unique solution
International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
@ IJTSRD | Unique Paper ID – IJTSRD42321 | Volume – 5 | Issue – 4 | May-June 2021 Page 553
� ≡ ����8��.
The congruence �� ≡ 3����8��, p an odd prime has a
unique solution
� ≡ 3����8��.
The congruence �� ≡ 2����3��, p an odd prime has a
unique solution
� ≡ 2����3��.
The congruence �� ≡ 5����8��, p an odd prime has a
unique solution
� ≡ 5����8��.
The congruence �� ≡ 7����8��, p an odd prime has a
unique solution
� ≡ 7����8��.
MERIT OF THE PAPER
The use of Chinese remainder theorem is needless.
Solutions can be obtained orally. This is the merit of the
paper.
REFERENCE
[1] Zuckerman H. S., Niven I., 2008, An Introduction to
the Theory of Numbers, Wiley India, Fifth Indian
edition, ISBN: 978-81-265-1811-1.
[2] David M Burton, 2012, Elementary Number Theory,
McGraw Hill education (Higher Education), Seventh
Indian Edition, New Dehli, India, ISBN: 978-1-25-
902576-1.
[3] Thomas Koshy, 2009, Elementary Number Theory
with Applications, Academic Press, Second Edition,
Indian print, New Dehli, India, ISBN: 978-81-312-
1859-4
[4] Roy B M, Formulation of a class of standard cubic
congruence modulo a positive prime integer
multiple of nine, ISSN: International Journal of
Recent Innovations in Academic Research (IJRIAR),
ISSN: 2635-3040, vol-02, Issue-05, Sept-18.
[5] Roy B M, Formulation of solutions of a class of
standard cubic congruence modulo �*+power of an
integer multiple of �*+ power of three, International
Journal of Recent Innovations in Academic
Research (IJRIAR), ISSN: 2635-3040, Vol-03, Issue-
01, Jan-19.
[6] Roy B M, Formulation of Two Special Classes of
Standard Cubic Congruence of Composite
Modulus—a power of three, International Journal of
Scientific Research and Engineering Development
(IJSRED), 2581-7175,Vol-02, Issue-03,May-19.
[7] Roy B M, Solving some special standard cubic
congruence modulo an odd prime multiplied by
eight, International Journal of Scientific Research
and Engineering Development(IJSRED), ISSN: 2581-
7175, Vol-04, Issue-01, Jan-21.
