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1. International Journal of Scientific Research and Engineering Development-– Volume 2 Issue 6, Nov- Dec 2019
Available at www.ijsred.com
ISSN : 2581-7175 ©IJSRED: All Rights are Reserved Page 120
RP-102: Formulation of Solutions of a Very Special Class
of Standard Quadratic Congruence of a Multiple of
Prime-power Modulus
Prof B M Roy
M. Sc. (Maths), Hon. Ph. D., Hon. D. Sc.
Head, department of Mathematics
Jagat Arts, Commerce & I H P Science College, Goregaon
Dist- GONDIA, M. S., INDIA. Pin: 441801
ABSTRACT
In this paper, the solutions of a very special class of standard quadratic congruence of a multiple of prime-
power modulus is considered forformulation and the solutions are formulated successfully. Such congruence
havelarge numbers of solutions. Solutions can be obtained easily in a short time. It is also possible to find the
solutions orally. Thus the formulation of solutions of the congruence is the merit of the paper. This formulation
made the study of quadratic congruence very interesting. A large numbers of solutions can be calculated
mentally. This is one more merit of the paper. Total number of solutions are double the prime in the modulus.
Key-words: Composite modulus,Formulation, Prime-power modulus, Quadratic congruence.
……………………………………………………………………………………………………………
INTRODUCTION
A standard quadratic congruence is a congruence of the type: ≡ ; m being a
Prime or Composite integer. The solutions are the values of that satisfy the congruence. If it is a standard
quadratic congruence of prime modulus, then it has exactly two solutions [1].But if it is a quadratic
congruence of composite modulus, then it may have more than two solutions [2]. Here, the author wishes to
formulate the solutions of the very special standard quadratic congruence of a multiple of prime-power
modulus of the type ≡ . ; p≥ 2, a positive prime integer and b any positive integer.
LITERATURE-REVIEW
In the literature of mathematics, a standard quadratic congruence of prime modulus is discussed prominently.
A little discussion is found on quadratic congruence of prime-power modulus.
It is found that if , = 1, then ≡ has exactly two solutions [2]. But no formulation is seen.
Earlier mathematicians (it seems) were not much interested in it. The author’s successful efforts opens the gate
of entry to the solutions of the congruence directly. He (the author) already formulated many standard
quadratic congruence of prime and composite modulus [3] to [12].
RESEARCH ARTICLE OPEN ACCESS

2. International Journal of Scientific Research and Engineering Development-– Volume 2 Issue 6, Nov- Dec 2019
Available at www.ijsred.com
ISSN : 2581-7175 ©IJSRED: All Rights are Reserved Page 121
NEED OF RESEARCH
Though the author formulated many standard quadratic congruence of prime and composite modulus, even he
found one more such special congruence yet remained to formulate. Here in this paper, the author considered
such congruence for formulation and his efforts are presented here. This is the need of the paper.
PROBLEM-STATEMENT
Here the problem is-
“To formulate the solutions of the special standard quadratic congruence:
≡ . ; , Positive integers, ≥ 2".
ANALYSIS & RESULTS
Consider the congruence ≡ . ; , positive integers, ≠ ; ≥ 2.
Let us consider that ≡ . ± . .
Then ≡ . ± (mod b.
≡ . , by binomial expansion formula.
Thus, ≡ . ± . is a solution of thecongruence:
≡ . ; , Positive integers, ≥ 2.
But, if we consider = , then ≡ . . ± .
≡ . ± .
≡ 0 ± ≡ ± .
Which is the same solution as for = 0.
Similarly, for higher values of k, the solutions repeats as for = 1, 2, 3, … . . , − 1 .
Therefore, all the required solutions are given by
≡ . ± . ; = 0, 1, 2, … … … − 1 .
These are 2 incongruent solutions for all values of k. The congruence has two solutions for every value of k
and k has p different values.
In particular, if = 1, "ℎ one get the solutions
≡ ± ; = 0, 1, 2, … … … − 1 .
ILLUSTRATIONS
Consider the congruence ≡ 49 1715 .
It can be written as ≡ 7 5.7(
.
It is of the type ≡ . ; )*"ℎ = 7, = 3, = 5.
The solutions are
≡ . ± . ; = 0, 1, 2, … … … − 1 .

3. International Journal of Scientific Research and Engineering Development-– Volume 2 Issue 6, Nov- Dec 2019
Available at www.ijsred.com
ISSN : 2581-7175 ©IJSRED: All Rights are Reserved Page 122
≡ 5.7(
± 7 5. 7(
; = 0, 1, 2, … … … … . . , 7 − 1.
≡ 245 ± 7 1715 ; = 0, 1, 2, 3, … … .6.
≡ 0 ± 7; 245 ± 7; 490 ± 7; 735 ± 7; 980 ± 7; 1225 ± 7; 1470 ± 7 1215
≡ 7, 1708;238, 252; 483, 497; 728, 742; 973, 987; 1218, 1232; 1463, 1477 1215 .
There are 14 solutions of the congruence.
Consider the congruence when = 1. *. . "ℎ - . / - ≡ 25 625 .
It is of the type ≡ 5 50
)*"ℎ = 5, = 4.
The solutions are ≡ ± ; = 0, 1, 2, … … … − 1 .
≡ 50
± 5 50
; = 0, 1, 2, … … … 5 − 1 .
≡ 125 ± 5 625 ; = 0,1, 2, 3,4.
≡ 0 ± 5; 125 ± 5; 250 ± 5; 375 ± 5; 500 ± 5 625 .
≡ 5, 620; 120, 130; 245, 255; 370, 380; 495, 505 625 .
There are 10 solutions of the congruence.
Consider the congruence ≡ 25 3.25 .
It is of the type ≡ 5 3.5 )*"ℎ = 5, = 2.
The solutions are ≡ . ± . ; = 0, 1, 2, … … … − 1 .
≡ 3. 5 ± 5 3. 5 ; = 0, 1, 2, … … … 5 − 1 .
≡ 15 ± 5 75 ; = 0,1, 2, 3,4.
≡ 0 ± 5; 15 ± 5; 30 ± 5; 45 ± 5; 60 ± 5 75 .
≡ 5, 70; 10, 20; 25, 35; 40, 50; 55, 65 75 .
There are 10 solutions of the congruence.
CONCLUSION
Therefore, it can be concluded that the congruence ≡ . ; ≥ 2, has 2p incongruent solutions
≡ . ± . ; = 0, 1, 2, … … . − 1 .
But if = 1, "ℎ "ℎ - . / - /- 1 " ≡ and also has 2p incongruent
Solutions given by ≡ ± ; = 0, 1, 2, … … . − 1 .
MERIT OF THE PAPER
Here, in this paper, a special type of standard quadratic congruence of prime-power modulus is formulated.
Formulation is the merit of the paper. It made the study of congruence very interesting.
REFERENCE
1. H S Zuckerman at el, 2008, An Introduction to The Theory of Numbers, fifth edition, Wiley student
edition, INDIA, ISBN: 978-81-265-1811-1.

4. International Journal of Scientific Research and Engineering Development-– Volume 2 Issue 6, Nov- Dec 2019
Available at www.ijsred.com
ISSN : 2581-7175 ©IJSRED: All Rights are Reserved Page 123
2. Thomas Koshy, 2009, “Elementary Number Theory with Applications”, 2/e Indian print, Academic
Press, ISBN: 978-81-312-1859-4.
3. Roy B M, 2018, A new method of finding solutions of a solvable standard quadratic congruence of
comparatively large prime modulus, International Journal of Advanced Research, Ideas and
Innovations in Technology (IJARIIT), ISSN:2454-132X, Vol-4, Issue-3,May-Jun-18.
4. Roy B M, 2018, Formulation of solutions of standard quadratic congruence of even composite
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Academic Research (IJRIAR), ISSN:2635-3040, Vol-2, Issue-2, Jun-18.
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and Innovations in Technology (IJARIIT),ISSN: 2454-132X, Vol-4, Issue-4,July-18.
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composite modulus as a product of a prime-power integer by two or four, International Journal for
Research Trends and Innovations(IJRTI), ISSN:2456-3315, Vol-3, Issue-5, May-18.
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Research Trends and Innovations(IJRTI), ISSN:2456-3315, Vol-3, Issue-5, May-18.
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of prime-power integer and eight, International Journal of Science & Engineering Development
Research (IJSDR),ISSN: 2455-2631, Vol-3, Issue-7, Jul-18.
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composite modulus, International Journal of Science & Engineering Development Research (IJSDR),
ISSN: 2455-2631, Vol-3, Issue-8, Jul-18.
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power Modulus, International Journal of Advanced Research, Ideas and Innovations in Technology
(IJARIIT), ISSN: 2454-132X, Vol-4, Issue-6, Dec-18.
11. Roy B M, 2019, Formulation of a Class of Solvable Standard Quadratic Congruence of Even
Composite Modulus, International Journal for Research Trends and Innovations (IJRTI), ISSN:2456-
3315, Vol-4, Issue-3, Mar-19.
12. Roy B M, 2019,Formulation of Some Classes of Solvable Standard Quadratic Congruence modulo a
Prime Integer - Multiple of Three & Ten, International Journal of Scientific Research and Engineering
Development ( IJSRED), ISSN: 2581-7175, Vol-2, Issue-2, Mar-19.