International Journal of Trend in Scientific Research and Development (IJTSRD)
Volume 5 Issue 4, May-June 2021 Available Online: www.ijtsrd.com e-ISSN:
ISSN: 2456 – 6470
RP-166:
166: Solving Some Special Standard Cubic Congruence of
Composite Modulus modulo a Multiple of an
n Odd Prime
Prof B M Roy
Head, Department of Mathematics, Jagat Arts, Commerce & I H P Science College, Goregaon,
Goregaon Maharashtra, India
How to cite this paper:
paper 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: 24562456
IJTSRD42321
6470, Volume-5
Volume
|
Issue-4,
4, June 2021,
pp.551-553,
553,
URL:
www.ijtsrd.com/papers/ijtsrd42321.pdf
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 solution
solu
is given in the problems. No
extra effort is necessary to find the solution.
KEYWORDS: Cubic Congruence, Composite Modulus, Unique Solution
Copyright © 2021
20
by author (s) and
International Journal of Trend in
Scientific Research and Development
Journal. This is an Open Access article
distributed
un
under
the terms of the
Creative Commons
Attribution License (CC BY 4.0)
(http://creativecommons.org/licenses/by/4.0
http://creativecommons.org/licenses/by/4.0)
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.
congruence of degree one and standard quadratic
congruence of prime and composite modulus
modul
are
remained in the part of study [1], [2], [3].Also some of the
author’s papers are seen [4], [5], [6].
Those solutions are present in the congruence itself. Here
is the list of those cubic Congruence
gruence in the problem
statement.
ANALYSIS & RESULTS
Consider the congruence:
odd prime.
PROBLEM-STATEMENT
“To find formula for solutions of the congruence:
≡
2 ,
It is easily seen that:
0
8 as
.
≡
Hence
3 ,
≡2
4 ,
≡3
Hence
6 ,
≡2
6 ,
≡3
6 ,
≡4
6 ,
≡5
6 ,
|
!
≡
Hence
3
Unique Paper ID – IJTSRD42321
42321
|
≡2
≡
3 . Here p is an
3
It is easily seen that:
1
.4 ≡ 0
4
|
.3 ≡
is a solution of the congruence.
≡2
2
.
3 . Here p is an
2 2
1 2" # 1
is a solution of the congruence.
Consider the congruence:
odd prime.
Volume – 5 | Issue – 4
.2 ≡
1 "#1
It is easily seen that: 2
.3 ≡ 0
3 as
."
1
is a solution of the congruence.
Consider the congruence:
odd prime.
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
@ IJTSRD
2 . Here p is an
It is easily seen that:
0
3 as
.
4 ,
≡
2
Consider the congruence:
odd prime.
3 ,
≡
≡
≡
≡
4 . Here p is an
!
as
May-June 2021
202
1
1
#
.
Page 551
International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
Hence
≡
4
is a solution of the congruence.
Consider the congruence:
odd prime.
≡3
Hence
3
3
4
8
≡
seen
1 ]
3 .8 ≡ 0
Hence
≡
!
≡3
that:
3
.8 ≡
3 9
!
≡
8
≡7
8.7
≡7
56 .
≡ 3.7
≡3
It has single solution
1
8 . Here p is an
It is of the type
( ℎ
≡3
8
≡ 3.7
8.7
≡ 21
56 .
≡ 5.7
It has single solution
5
5 25
!
1
8 .
8
is a solution of the congruence.
≡7
It is easily seen that:
7 [48 ! # ! 1 ]
7
7 .8 ≡ 0
8 . Here p is an
7 49
!
1
8
is a solution of the congruence.
ILLUSTRATIONS
Example-1:Consider the congruence
It can be written as
≡7
≡
It has single solution
It can be written as
It has single solution
≡7
2.7
≡7
14 .
≡ 35
56 .
≡ 7.7
3
≡7
3.7
≡7
21 .
The congruence
unique solution
( ℎ
8
≡ 7.7
8.7
≡ 49
56 .
≡
( ℎ
≡
The congruence
unique solution
The congruence
unique solution
3 , p an odd prime has a
≡
3
≡3
Unique Paper ID – IJTSRD42321
|
Volume – 5 | Issue – 4
|
≡
.
4 , p an odd prime has a
4 .
4 , p an odd prime has a
≡3
The congruence
unique solution
2 , p an odd
3 .
≡
3
7.
2 .
≡2
7.
56 .
3 , p an odd prime has a
≡2
3.7 .
3
8
≡7
7.
21 .
≡ 49
8.7 .
≡
≡ 14
7.
CONCLUSION
It can be concluded from this discussion that the standard
cubic congruence considered, each has a single solutions.
The congruence
unique solution
21 .
≡
≡2
7.
( ℎ
≡ 2.7
8.7
It is found that the congruence
prime has a unique solution
3.7 .
3
≡2
It has single solution
≡7
≡7
≡ 5.7
≡
2
Consider the congruence
It can be written as
( ℎ
≡
≡
14 .
2.7 .
2
Consider the congruence
≡7
( ℎ
8
It has single solution
8 .
≡7
8
≡7
56 .
8.7 .
≡5
It can be written as
56 .
7.
≡ 35
Example-4: Consider the congruence
It is of the type
7
≡ 21
8.7 .
8
≡5
56 .
7.
Example-2: Consider the congruence
It is of the type
is a solution of the congruence.
Consider the congruence:
odd prime.
|
It has single solution
( ℎ
It can be written as
≡5
@ IJTSRD
8
8
5 .8 ≡ 0
It is of the type
≡
≡7
8.7 .
Example-3: Consider the congruence
5
It is of the type
≡7
It can be written as
8 .
It is easily seen that:
5 [24 ! # ! 1 ]
It is of the type
1
8 . Here p is an
3
≡5
Hence
21 .
It can be written as
Consider the congruence:
odd prime.
Hence
≡ 14
8 . Here p is an
is a solution of the congruence
Consider the congruence:
odd prime.
It is easily
3 [8 ! # !
1
is a solution of the congruence.
It is easily seen that:
8 .
0
8 as ! ≡ 1
≡
!
It is of the type
Consider the congruence:
odd prime.
Hence
3 9
4 .
≡3
3.7
Example-1: Consider the congruence
It is easily seen that:
3 [4 ! # 4 ! # ! 1 ]
3 .4 ≡ 0
4 . Here p is an
≡ 2.7
4
.
8 , p an odd prime has a
May-June 2021
Page 552
International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
≡
The congruence
unique solution
≡3
≡3
The congruence
unique solution
≡2
≡2
The congruence
unique solution
≡5
≡5
The congruence
unique solution
≡7
≡7
8 .
8 , p an odd prime has a
8
3
[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, Issue01, 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: 25817175, Vol-04, Issue-01, Jan-21.
.
8 , p an odd prime has a
8
.
8 , p an odd prime has a
8
.
David M Burton, 2012, Elementary Number Theory,
McGraw Hill education (Higher Education), Seventh
|
Thomas Koshy, 2009, Elementary Number Theory
with Applications, Academic Press, Second Edition,
Indian print, New Dehli, India, ISBN: 978-81-3121859-4
3 , p an odd prime has a
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.
@ IJTSRD
[3]
.
MERIT OF THE PAPER
The use of Chinese remainder theorem is needless.
Solutions can be obtained orally. This is the merit of the
paper.
[2]
Indian Edition, New Dehli, India, ISBN: 978-1-25902576-1.
Unique Paper ID – IJTSRD42321
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Volume – 5 | Issue – 4
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May-June 2021
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