Rp-26: Formulation of Solutions of Standard Cubic Congruence of Even Composite Modulus- An Eighth Multiple of Nth Power of Three

International Journal of Scientific Research and Engineering Development-– Volume 3 Issue 4, July-Aug 2020
Available at www.ijsred.com
ISSN : 2581-7175 ©IJSRED:All Rights are Reserved Page 202
Rp-26: Formulation of Solutions of Standard Cubic Congruence of
Even Composite Modulus- An Eighth Multiple of Nth Power of Three
Prof. B. M. Roy
Head, Dept. of Mathematics
Jagat Arts, Commerce & I H P Science College, Goregaon (Gondia).
Dist. - GONDIA, M. S., India, Pin-441801
(Affiliated to R T M Nagpur University)
ABSTRACT
Here in this paper, the finding of solutions of a standard cubic congruence of even composite
modulus-an eighth multiple of nth power of three, is formulated. The formulation is establishedfor its
solutions and the established formula is tested and verified using different numerical examples.
Formulation is the merit of the paper. It is found that the said congruence has exactly three / twelve
solutions as per the case.
Keywords: Composite modulus; Chinese Remainder Theorem,Cubic congruence, Cubic residue.
INTRODUCTION
Here, the author considered a standard cubic congruence of even composite modulus of the type:
𝑥 ≡ 𝑎 ( 𝑚𝑜𝑑 8. 3 ) for its formulation of solutions. The author already has formulated many
different types of standard cubic congruence of composite modulus[3], [4], [5], [6].Here, he
found one more such type of standard cubic congruence of compositemodulus yet unformulated
and hence he considered it for its formulation of its solutions.
RESEARCH ARTICLE OPEN ACCESS

PROBLEM STATEMENT
The problem is:
“To formulate of the solutions of the standard cubic congruence of even composite modulus:
𝑥 ≡ 𝑎 (𝑚𝑜𝑑 8. 3 ), n is a positive integer in two cases:
Case-I: when 𝑎 is odd positive integer;
Case-II: when 𝑎 is even positive integer.
Case-III: When 𝑎 = 3𝑚, 𝑚 𝑏𝑒𝑖𝑛𝑔 an integer.
LITERATURE-REVIEW
The standard cubic congruence under consideration is not formulated earlier. No formulation is found
in the literature of mathematics except Thomas Koshy [1] and Zuckerman [2]. They only defined the
cubic congruence and cubic residue and stated no method and no formulation for its solutions. But it
is seen that it can be solved using the Chinese Remainder Theorem (C R T).
EXISTED METHOD
In existed method, the solutions of such congruence are obtained by separatingthe congruence of
composite modulus into individual congruence as here: 𝑥 ≡ 𝑎 (𝑚𝑜𝑑 8)&𝑥 ≡ 𝑎 (𝑚𝑜𝑑 3 ), and
solving these two individual congruence separately, required solutions are obtained using Chinese
Remainder Theorem. The use of Chinese Remainder Theorem [1] is a time-consuming procedure.
This is the demerit of the existed method.
ANALYSIS & RESULT
Consider the congruence under consideration: 𝑥 ≡ 𝑎 ( 𝑚𝑜𝑑 8. 3 ).
Case-I: Let 𝑎 be an odd positive integer.

For its solutions, consider 𝑥 ≡ 8. 3 𝑘 ± 𝑎 ( 𝑚𝑜𝑑 8. 3 ).
Then 𝑥 ≡ (8. 3 𝑘) + 3. ( 8. 3 𝑘) . 𝑎 + 3.8. 3 𝑘. 𝑎 + 𝑎 ( 𝑚𝑜𝑑 8. 3 ).
≡ (8. 3 𝑘) + 3. ( 8. 3 𝑘) . 𝑎 + 8. 3 𝑘. 𝑎 + 𝑎 (𝑚𝑜𝑑 8. 3 )
≡ 8. 3 𝑘(8 . 3 𝑘 + 8. 3 𝑘𝑎 + 𝑎 ) + 𝑎 (𝑚𝑜𝑑 8. 3 )
≡ 8. 3 𝑘( 𝑡) + 𝑎 (𝑚𝑜𝑑 8. 3 ), if 𝑎 is odd positive integer.
≡ 𝑎 (𝑚𝑜𝑑 8. 3 )
Therefore, 𝑥 ≡ 8. 3 𝑘 + 𝑎 ( 𝑚𝑜𝑑 8. 3 ) is a solution. But for 𝑘 = 3, the solution is the same as for
𝑘 = 0.
The solutions repeats as for 𝑘 = 1, 2 for higher values of k such as 𝑘 = 4, … ….
Thus, 𝑥 ≡ 8. 3 𝑘 ± 𝑎 ( 𝑚𝑜𝑑 8. 3 ); 𝑘 = 0, 1, 2 gives the required solutions of the congruence.
These are the three solutions of the congruence.
Case-II: Let 𝑎 be an even positive integer.
For its solutions, consider 𝑥 ≡ 2. 3 𝑘 ± 𝑎 ( 𝑚𝑜𝑑 8. 3 ).
Then 𝑥 ≡ (2. 3 𝑘) + 3. ( 2. 3 𝑘) . 𝑎 + 3.2. 3 𝑘. 𝑎 + 𝑎 ( 𝑚𝑜𝑑 8. 3 ).
≡ (2. 3 𝑘) + 3. ( 2. 3 𝑘) . 𝑎 + 2. 3 𝑘. 𝑎 + 𝑎 (𝑚𝑜𝑑 8. 3 )
≡ 2. 3 𝑘(2 . 3 𝑘 + 2. 3 𝑘𝑎 + 𝑎 ) + 𝑎 (𝑚𝑜𝑑 8. 3 )
≡ 2. 3 𝑘( 4𝑡) + 𝑎 (𝑚𝑜𝑑 8. 3 ), if 𝑎 is even positive integer.
≡ 𝑎 (𝑚𝑜𝑑 8. 3 )
Therefore, 𝑥 ≡ 2. 3 𝑘 + 𝑎 ( 𝑚𝑜𝑑 8. 3 ) are thesolutions. But for 𝑘 = 12, the solution is the same
as for 𝑘 = 0.
The solutions repeats as for 𝑘 = 1, 2, ….for higher values of k such as 𝑘 = 13, 14, … ….
Thus, 𝑥 ≡ 2. 3 𝑘 ± 𝑎 ( 𝑚𝑜𝑑 8. 3 ); 𝑘 = 0, 1, 2, … … . .11 gives the required solutions of the
congruence. These are the twelve solutions of the congruence.
Case-III: Let 𝑎 = 3𝑚; 𝑚 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟.
For the solutions, consider 𝑥 ≡ 8. 3 𝑘 ± 𝑎 (𝑚𝑜𝑑 8. 3 ).
Then 𝑥 ≡ (8. 3 𝑘) + 3. ( 8. 3 𝑘) . 𝑎 + 3.8. 3 𝑘. 𝑎 + 𝑎 ( 𝑚𝑜𝑑 8. 3 ).
≡ (8. 3 𝑘) + 3. ( 8. 3 𝑘) . 𝑎 + 8. 3 𝑘. 𝑎 + 𝑎 (𝑚𝑜𝑑 8. 3 )

≡ 8. 3 𝑘(8 . 3 𝑘 + 8. 3 𝑘𝑎 + 𝑎 ) + 𝑎 (𝑚𝑜𝑑 8. 3 )
≡ 8. 3 𝑘( 𝑡) + 𝑎 (𝑚𝑜𝑑 8. 3 ), if 𝑎 = 3𝑡, 𝑡 = 1, 2 is odd positive integer.
≡ 𝑎 (𝑚𝑜𝑑 8. 3 )
Therefore, 𝑥 ≡ 8. 3 𝑘 + 𝑎 (𝑚𝑜𝑑 8. 3 ) is a solution. But for 𝑘 = 9, the solution is the same as for
𝑘 = 0.
The solutions repeats as for 𝑘 = 1, 2 ,3 for higher values of k such as 𝑘 = 10,11, 12 … ….
Thus, 𝑥 ≡ 8. 3 𝑘 ± 𝑎 ( 𝑚𝑜𝑑 8. 3 ); 𝑘 = 0, 1, 2,3, … … . ,8,9 gives the required solutions of the
congruence. These are the nine solutions of the congruence.
ILLUSTRATIONS
Example-1: Let us consider an example𝑥 ≡ 125 ( 𝑚𝑜𝑑 648).
It can be written as: 𝑥 ≡ 5 (𝑚𝑜𝑑 8.3 )
Here, 648 = 8.81 = 8.3
So, the congruence is of the type 𝑥 ≡ 𝑎 ( 𝑚𝑜𝑑 8.3 ) and has only three solutions.
The solutions are given by 𝑥 ≡ 8.3 𝑘 + 𝑎 ( 𝑚𝑜𝑑 8.3 ); 𝑘 = 0, 1, 2.
≡ 8.27𝑘 + 5 ( 𝑚𝑜𝑑 648 ); 𝑘 = 0, 1, 2.
≡ 216𝑘 + 5 (𝑚𝑜𝑑 648)
≡ 0 + 5; 216 + 5; 432 + 5 ( 𝑚𝑜𝑑 648).
≡5, 221, 437 (mod 648).
It can be written as: 𝑥 ≡ 2 ( 𝑚𝑜𝑑 8.3 )
Here, 216 = 8.27 = 8.3
So, the congruence is of the type 𝑥 ≡ 𝑎 ( 𝑚𝑜𝑑 8.3 ) and has only twelve solutions.
The solutions are given by
𝑥 ≡ 2.3 𝑘 + 𝑎 (𝑚𝑜𝑑 8.3 ); 𝑘 = 0, 1, 2, 3, 4 …… … 10, 11.
≡ 2.9𝑘 + 2 ( 𝑚𝑜𝑑 216 ); 𝑘 = 0, 1, 2 … … … … . , 10, 11.
≡ 18𝑘 + 2 (𝑚𝑜𝑑 216)
≡ 0 + 2; 18 + 2; 36 + 2; 54 + 2; 72 + 2; 90 + 2; 108 + 2; 126 + 2; 144 + 2; 162 + 2; 180
+ 2; 198 + 2 ( 𝑚𝑜𝑑 216).
≡2, 20, 38, 56, 74, 92, 110, 128, 146, 164, 182, 200 (mod 216).

Here, 216 = 8.27 = 8.3
So, the congruence is of the type 𝑥 ≡ 𝑎 (𝑚𝑜𝑑 8.3 ) and has only nine solutions.
𝑥 ≡ 8.3 𝑘 + 𝑎 ( 𝑚𝑜𝑑 8.3 ); 𝑘 = 0, 1, 2, … … . .7, 8.
≡ 8.3𝑘 + 3 (𝑚𝑜𝑑 216 ); 𝑘 = 0, 1, 2, … … … … 7, 8.
≡ 24𝑘 + 3 (𝑚𝑜𝑑 216)
≡ 0 + 3; 24 + 3; 48 + 3; 72 + 3; 96 + 3; 120 + 3;
144 + 3; 168 + 3; 192 + 3 ( 𝑚𝑜𝑑 216).
≡3, 27, 51, 75, 99, 123, 147, 171, 195 (mod 216).
Here, 648 = 8.81 = 8.3
So, the congruence is of the type 𝑥 ≡ 𝑎 ( 𝑚𝑜𝑑 8.3 ) and has only nine solutions.
𝑥 ≡ 8.3 𝑘 + 𝑎 (𝑚𝑜𝑑 8.3 ); 𝑘 = 0, 1, 2, … … . .7, 8.
≡ 8.9𝑘 + 3 ( 𝑚𝑜𝑑 648 ); 𝑘 = 0, 1, 2, … … … … 7, 8.
≡ 72𝑘 + 3 (𝑚𝑜𝑑 648)
≡ 0 + 6; 72 + 6; 144 + 6; 216 + 6; 288 + 6; 360 + 6
432 + 6; 504 + 6; 576 + 6 ( 𝑚𝑜𝑑 648).
≡6, 78, 150, 222, 294, 366, 438, 510, 582 (mod 648).
CONCLUSION
Thestandard cubic congruence under consideration has exactly three solutions given by
𝑥 ≡ 8. 3 𝑘 ± 𝑎 ( 𝑚𝑜𝑑 8. 3 ); 𝑘 = 0, 1, 2if𝑎 is an odd positive integer.
The standard cubic congruence under consideration has exactly twelve solutions given by
𝑥 ≡ 2. 3 𝑘 ± 𝑎 ( 𝑚𝑜𝑑 8. 3 ); 𝑘 = 0, 1, 2, … … . . , 11if𝑎 is an even positive integer.
But if 𝑎 = 3𝑡, a multiple of three, then it has nine solutions given by
𝑥 ≡ 8. 3 𝑘 ± 𝑎 ( 𝑚𝑜𝑑 8. 3 ); 𝑘 = 0, 1, 2, 3, 4, 5, 6, 7, 8.

Therefore, the standard cubic congruence of composite modulus is formulated and discussed with
illustration completely.
MERIT OF THE PAPER
In this paper, the standard cubic congruence under consideration is formulated. Using the
formulation, the solutions can be obtained easily. This is the merit of the paper.
REFERENCE
[1] Koshy, Thomas, Elementary Number Theory with Applications; 2/e; Academic press. ISBN: 978-
81-312-1859-4.
[2] Niven, I., Zuckerman H S.; Montgomery H L, An Introduction to the Theory of Numbers; 5/e;
WSE, ISBN: 978-81-265-1811-1.
[3] Roy B M, Formulation of solutions of a standard cubic congruence of composite modulus- twice
a prime multiple of power of three, International Journal of Scientific Research and Engineering
Development (IJSRED), ISSN: 2581-7175, Vol-03, Issue-02, Mar-20.
[4] Roy B M, Formulation of solutions of three very special standard cubic congruence of composite
modulus,International Journal for Research Trends and Innovation (IJRTI), ISSN: 2456--3315, Vol-
05, Issue-02, Feb-20.
[5] Roy B M, Formulation of solutions of standard cubic congruence of even composite modulus-an
even multiple of power of three, International Journal for Research Trends and Innovation(IJRTI),
ISSN: 2456-3315, Vol-05, Issue-04, April-20.
[6] Roy B M, Formulation of solutions of standard cubic congruence of composite modulus-a
multiple of four & power of three, International Journal of Trend in Scientific Research and
development, (IJTSRD), ISSN: 2456-6470, Vol-04, Issue-04, May-Jun-20.
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Rp-26: Formulation of Solutions of Standard Cubic Congruence of Even Composite Modulus- An Eighth Multiple of Nth Power of Three

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Rp-26: Formulation of Solutions of Standard Cubic Congruence of Even Composite Modulus- An Eighth Multiple of Nth Power of Three