RP 58 Formulation of Solutions of Standard Cubic Congruence of Composite Modulus A Product of Eight...

International Journal of Trend in Scientific Research and Development (IJTSRD)Volume 4 Issue 4, June 2020

@ IJTSRD | Unique Paper ID – IJTSRD3156

RP-58: Formulation o

of Composite Modulus

Odd Prime & Nth Power

Head, Department of Mathematics

Goregaon

(Affiliated to R T M Nagpur University)

ABSTRACT

In this paper, a standard cubic congruence of even composite modulus

product eight multiple of an odd prime & nth power of three, is considered

for solutions. The congruence is studied and

established. The formulation is done in two cases with different formula of

solutions. Using the formula, the finding of solutions becomes easy and

time-saving. Formulation is the merit of the paper.

KEYWORDS: Cubic Congruence, Composite Modulus, Cubic

Formulation

INTRODUCTION:

A cubic congruence is seldom studied in Number Theory. The author started to study the topic and formulated many

standard cubic congruence of composite modulus. Here is one of such standard cu

not formulated earlier. Hence, it is considered for study and formulation of its solutions.

PROBLEM-STATEMENT

“To formulate the solutions of the standard cubic congruence:

Case-I : � ≡ 2��3�� 3� is � positive odd integer;

Case-II : � ≡ 2��3�� 3�isan

Case-III : � � 3�; ��;

Case-IV : � � 3�; ��

LITERATURE REVIEW

The congruence under consideration has not been formulated earlier by mathematicians.

Actually no detailed study is found in the literature. Zukerman [1] and Thomas Koshy [2] had given a

small start but no formulation is found in the literature. Only the author’s formulations of some of the standard cubic

congruence of prime and composite modulus are found [3], [4], [5], [6].

ANALYSIS & RESULTS

Consider the congruence � ≡ � ��3!

Case-I: Also, let � 3� be an odd positive integer.

International Journal of Trend in Scientific Research and Development (IJTSRD)2020 Available Online: www.ijtsrd.com e-ISSN: 2456

31566 | Volume – 4 | Issue – 4 | May-June 2020

of Solutions of Standard Cubic Congruence

f Composite Modulus - A Product of Eight-Multiple

Odd Prime & Nth Power of Three

Prof B M Roy

Head, Department of Mathematics, Jagat Arts, Commerce & I H P Science College,

Goregaon, Gondia, Maharashtra, India

(Affiliated to R T M Nagpur University)

In this paper, a standard cubic congruence of even composite modulus- a

product eight multiple of an odd prime & nth power of three, is considered

udied and formulation of its solution is

established. The formulation is done in two cases with different formula of

solutions. Using the formula, the finding of solutions becomes easy and

saving. Formulation is the merit of the paper.

bic Congruence, Composite Modulus, Cubic-Residue,

How to cite this paper

"RP-58: Formulation of Solutions of

Standard Cubic Congruence of

Composite Modulus

Multiple of an Odd Prime &

Three" Published in

International

Journal of Trend in

Scientific Research

and Development

(ijtsrd), ISSN: 2456

6470, Volume

Issue-4, June 2020,

pp.1443-1446, URL:

www.ijtsrd.com/papers/ijtsrd31566.pdf

Copyright © 20

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.

(http://creativecommons.org/licenses/

by/4.0)


standard cubic congruence of composite modulus. Here is one of such standard cubic congruence of composite modulus,

not formulated earlier. Hence, it is considered for study and formulation of its solutions.

“To formulate the solutions of the standard cubic congruence: � ≡ ��3!. 8�, � being an odd prime

positive odd integer;

anevenpositiveinteger.

”.

The congruence under consideration has not been formulated earlier by mathematicians.

Actually no detailed study is found in the literature. Zukerman [1] and Thomas Koshy [2] had given a


congruence of prime and composite modulus are found [3], [4], [5], [6].

! . 8�. ,�� ≡ 2��3.

be an odd positive integer.

International Journal of Trend in Scientific Research and Development (IJTSRD)

ISSN: 2456 – 6470

2020 443

of Standard Cubic Congruence

Multiple of an

f Three

Commerce & I H P Science College,

How to cite this paper: Prof B M Roy

58: Formulation of Solutions of

Standard Cubic Congruence of

Composite Modulus - A Product of Eight-

Multiple of an Odd Prime & Nth Power of

Three" Published in

International

Journal of Trend in

Scientific Research

and Development

(ijtsrd), ISSN: 2456-

6470, Volume-4 |

4, June 2020,

1446, URL:

www.ijtsrd.com/papers/ijtsrd31566.pdf

Copyright © 2020 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/


bic congruence of composite modulus,

being an odd prime

Actually no detailed study is found in the literature. Zukerman [1] and Thomas Koshy [2] had given a very


IJTSRD31566

International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470

@ IJTSRD | Unique Paper ID – IJTSRD31566 | Volume – 4 | Issue – 4 | May-June 2020 Page 1444

Then for � ≡ 3!-.. 8�/ + ��3!. 8�,

� ≡ �3!-.. 8�/ + � ��3!. 8�

≡ �3!-.. 8�/ + 3. �3!-.8�/1. � + 3. 3!-.. 8�/. �1 + � ��3!. 8�

≡ � + 3!. 8�/{�1 + 3!-.. 8�/. � + 31!- . �8�/1}��3! . 8�

≡ � ��3!. 8�.

Therefore, it is the solution of the congruence.

But for / = 3,4, 5 the solution is the same as for / = 0, 1, 2.

Thus is can be said that the solutions of the congruence are given by

� ≡ 3!-.. 8�/ + ��3!. 8�, / = 0, 1, 2.

Case-II:

Also, let � ≠ 3� be an even positive integer.

Then for � ≡ 3!-.. 2�/ + ��3!. 8�,

� ≡ �3!-.. 2�/ + � ��3!. 8�

≡ �3!-.. 2�/ + 3. �3!-.. 2�/1. � + 3. 3!-.. 2�/. �1 + � ��3!. 8�

≡ � + 3!. 8�/{…………………………… }��3! . 8�

≡ � ��3!. 8�, if �.is even positive integer.

Therefore, it is the solution of the congruence.

But for / = 12, 13, 14, …… the solutions are the same as for / = 0, 1, 2, …….

Thus it can be said that the solutions of the congruence are given by

� ≡ 3!-.. 2�/ + ��3!. 8�, / = 0, 1, 2, 3, ……… . .10, 11.

Case-III: let � = 3�; � = 1, 3, 5……….

Consider x ≡ 8p. 3!-1k + 3��mod8p. 3=,

Then x ≡ �8p. 3=-1k + 3� �mod8p. 3=

≡ �8p. 3=-1k + 3. �8p. 3=-1k1. 3� + 3.8p. 3=-1k. �3�1 + �3� �mod8p. 3=

≡ 8p. 3=k{64�1/131!-? + 8�/� + 3�} + �3� �mod8p. 3=

≡ �3� �mod8p. 3=, for a for any positive integer.

Thus, x ≡ 8p. 3=-1k + 3��mod8p. 3= satisfies the said congruence.

Hence, it is a solution of the congruence.

But for k = 9, thesolutionbecomesx ≡ 2. 3=-1. 9 + 3�mod8p. 3=

≡ 2. 3= + 3�mod8p. 3=

≡ 3�mod8p. 3=

Which is the same as for the solution for k = 0.

For k = 10, 11, ……….the solutions are also the same as for k = 1, 2…… ..

Thus, the congruence has exactly nine incongruent solutions

x ≡ 8p. 3=-1k + 3�mod8p. 3=, foranyintegera, k = 0, 1, 2, 3, 4, 5, 6, 7, 8.

Case-IV: But if� = 3�, � = 2, 4, … .., then for x ≡ 4p. 3=-1k + 3��mod8p. 3=,

� ≡ �4p. 3=-1k + 3� ��8p. 3=

≡ �4p. 3=-1k + 3. �4p. 3=-1k1. 3� + 3.4p. 3=-1k. �3�1 + �3� ��8p. 3=

≡ �3� + 4�. 3!/�41. 31!-I. �1/1 + 4�/� + 3�1��8p. 3=

≡ �3� ��8p. 3=�J�is even.

Therefore, x ≡ 4p. 3=-1k + 3��mod8p. 3= satisfies the cubic congruence and hence it is a solution of it.

But for / = 18 = 2.31, the solution reduces to x ≡ 4p. 3=-1. 2. 31 + 3��mod8p. 3=

≡ 8�. 3! + 3��8�. 3!

≡ 3��8�. 3!

Thus, it is the same solution as for / = 0.

Similarly, it is also seen that for / = 19, 20…… ., the solutions repeat as for / = 1, 2, … ..

Therefore, the congruence has eighteen solutions: x ≡ 4. 3=-1k + 3��mod8p. 3=;



/ =, 1, 2, ……… . . , 17.

Sometimes, in the cubic congruence, the integer a may not be a perfect cube. The readers have to make it so, by adding

multiples of the modulus [1].

ILLUSTRATIONS

Example-1: Consider the congruence: � ≡ 1 ��31. 8.5

It is of the type

� ≡ � ��3! . 8�K��ℎ� = 1, ��L��, � = 5 ≡ 2��3.

Such types of congruence has exactly three solutions given by

� ≡ 3!-.. 8�/ + ��3!. 8�

≡ 3.. 8.5/ + 1��31. 8.5

≡ 120/ + 1��360; / = 0, 1, 2.

≡ 1, 121, 241��360.

Example-2: Consider the congruence: � ≡ 4 ��31. 8.11

It is of the type

� ≡ � ��3! . 8. �K��ℎ� = 4, ��L��, � = 11 ≡ 2��3.

Such types of congruence has exactly twelve solutions given by

� ≡ 3!-.. 2�/ + ��3!. 8. �

≡ 3.. 2.11/ + 4��31. 8.11

≡ 66/ + 4��396; / = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11.

≡ 4, 70,136, 202, 268, 334,466, 400, 532, 598, 664, 730��792.

Example-3: Consider the

congruence:� ≡ 216��1080

It can be written as � ≡ 6 ��3?. 8.5

It is of the type

� ≡ � ��3! . 8�K��ℎ� = 6, ��L��, � = 5 ≡ 2��3.

Such types of congruence has exactly eighteen solutions given by

� ≡ 3!-1. 4�/ + ��3!. 8�

≡ 3.. 4.5/ + 6��3 . 8.5

≡ 60/ + 6��1080; / = 0, 1, 2, 3, 4, 5, ……… . . , 16, 17.

≡ 4, 66, 126, 186, 246, 306, 366, 426, 486, 546, 606, 666, 726,

786, 846, 906, 966, 1026��1080.

Example-4: Consider the congruence:� ≡ 729��2376

It can be written as � ≡ 9 ��3 . 8.11

It is of the type

� ≡ � ��3! . 8. �K��ℎ� = 9, ��L��, � = 11 ≡ 2��3.

Such types of congruence has exactly nine solutions given by

� ≡ 3!-1. 8�/ + ��3!. 8. �

≡ 3.. 8.11/ + 9��3 . 8.11

≡ 264/ + 9��2376; / = 0, 1, 2, 3, 4, 5, 6, 7, 8.

≡ 9, 273, 537, 801, 1065, 1329, 1593, 1857, 2121��2376.

CONCLUSION

Therefore, it is concluded that the cubic congruence under consideration:

� ≡ � ��3! . 8�; p≡ 2��3 has exactly three solutions, if � is an odd positive integer, given by � ≡ 3!-.. 8�/ +

��3!. 8�; / = 0, 1, 2;

while the congruence has exactly twelve solutions if � is an even positive integer, given by

� ≡ 3!-.. 2�/ + ��3! . 8�; / = 0, 1, 2, …… . . , 11.

If � = 3�, ��ℎ��, the congruence has exactly nine solutions given by

� ≡ 3!-1. 8�/ + ��3!. 8. �; / = 0, 1, 2, ……… .8.

If � = 3�, ��ℎ��, the congruence has exactly eighteen solutions given by

� ≡ 3!-1. 4�/ + ��3! . 8. �; / = 0, 1, 2, ……… .17.



REFERENCE

[1] Zuckerman H. S., Niven I., Montgomery H. L., “An

Introduction to The Theory of Numbers”, 5/e, Wiley

India (Pvt) Ltd, Page No. 136-136, Exercise-18, ISBN:

978-81-265-1811-1. (1960: Reprint 2008).

[2] Thomas Koshy, “Elementary Number Theory with

Applications”, 2/e (Indian print), Academic Press,

Page No. 548, supplementary Exercises No.4, and

ISBN: 978-81-312-1859-4, (2009).

[3] Roy B M, Formulation of solutions of a class of

standard cubic congruence modulo nth power of three,

International Journal for Research Under Literal

Access(IJRULA), Vol. 01, Issue-03,(Aug- 2018), Page

No. 254-256.

[4] Roy B. M., Formulation of solutions of a class of

standard cubic congruence modulo a positive prime

integer multiple of Nine, International Journal of

Recent Innovations In Academic research (IJRIAR),

ISSN: 2635-3040, Vol. 02, Issiue-5, Sept-2018.

[5] Roy B. M., Formulation of a class of standard Cubic

Congruence of even composite modulus, International

Journal of Advance Research, Ideas and Innovations

in Technology (IJARIIT), ISSN: 2454-132X, Vol-5,

Issue-1, Jan-Feb, 2019.

[6] Roy B M, Formulation of standard cubic congruence of

even composite modulus, International Journal of

Research and Analytical Reviews (IJRAR), ISSN:

2348-1296, Vol-06, Issue-02, Sep-19.

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