This study presents a novel formulation for solving standard quadratic congruences of composite modulus specifically involving products of two powered odd primes. The author highlights the lack of prior formulations in existing literature and demonstrates a method to find solutions through case studies. The paper emphasizes the significance of this contribution in the field of mathematical congruences.

1. International Journal of Trend in Scientific Research and Development (IJTSRD)
Volume 4 Issue 4, June 2020
@ IJTSRD | Unique Paper ID – IJTSRD30964
RP-129: Formulation
Quadratic Congruence
a Product
M. Sc. (maths); Ph. D. (Hon); D. Sc. (Hon)
Jagat Arts, Commerce & I H P Science College, Goregaon
ABSTRACT
In this current study, a very special type of standard quadratic congruence
of composite modulus- a product of two powered odd primes is considered
for finding solutions. It is studied and solved in different
effort made an easy way to find the solutions of the congruence under
consideration. Such types of congruence are not formulated earlier. First
time the author’s effort came in existence. This is the merit of the paper.
KEYWORDS: Composite modulus, CRT method, Legendre’s symbol, Quadratic
congruence
INTRODUCTION
The author formulated many standard quadratic
congruence of prime and composite modulus.
Here he found one more standard quadratic congruence of
composite modulus- a product of two powered primes yet
remained unformulated. He consider the problem for his
study and found a formulation for its solutions. His efforts
are presented in this current paper.
PROBLEM-STATEMENT
In this paper, the problem is:
“To find the solutions of the standard quadratic
congruence of composite modulus of the type
‫ݔ‬ଶ
≡ ܾ ሺ݉‫݌ ݀݋‬௡
‫ݍ‬௠ሻ; ‫, ݌‬ ‫,ݏ݁݉݅ݎ݌ ݀݀݋ ݁ݎܽ ݍ‬
ܽଶ
, ‫݌‬ଶ
, ‫ݍ‬ଶ
, ሺ݉‫݌‬ሻଶ
‫ ݎ݋‬ሺ݉‫ݍ‬ሻଶ
LITERATURE-REVIEW
In the literature of mathematics, a standard quadratic
congruence of prime modulus of discussed prominently.
But a very little material of solving standard quadratic
congruence of composite modulus is found. Only a method
known as Chinese Remainder Theorem (CRT) can be used
[3]. The author’s formulations of solutions of standard
quadratic congruence of composite modulus are found [4],
…….[8].
International Journal of Trend in Scientific Research and Development (IJTSRD)
2020 Available Online: www.ijtsrd.com e-ISSN: 2456
30964 | Volume – 4 | Issue – 4 | May-June 2020
129: Formulation of a Special Type of Standard
Congruence of Composite Modulus
Product of Two Powered Odd Primes
Prof B M Roy
M. Sc. (maths); Ph. D. (Hon); D. Sc. (Hon), Head, Department of Mathematics
Jagat Arts, Commerce & I H P Science College, Goregaon, Maharashtra
In this current study, a very special type of standard quadratic congruence
a product of two powered odd primes is considered
for finding solutions. It is studied and solved in different cases. The author’s
effort made an easy way to find the solutions of the congruence under
consideration. Such types of congruence are not formulated earlier. First
time the author’s effort came in existence. This is the merit of the paper.
posite modulus, CRT method, Legendre’s symbol, Quadratic
How to cite this paper
"RP-129: Formulation of a Special Type
of Standard Quadratic Congruence of
Composite Modulus
Powered Odd
Primes" Published
in Interna
Journal of Trend in
Scientific Research
and Development
(ijtsrd), ISSN: 2456
6470, Volume
Issue-4, June 2020, pp.326
www.ijtsrd.com/papers/ijtsrd30964.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 4.0)
(http://creativecommons.org/licenses/
by/4.0)
The author formulated many standard quadratic
congruence of prime and composite modulus.
Here he found one more standard quadratic congruence of
a product of two powered primes yet
emained unformulated. He consider the problem for his
study and found a formulation for its solutions. His efforts
“To find the solutions of the standard quadratic
congruence of composite modulus of the type
, ݂݅ ܾ ൌ
In the literature of mathematics, a standard quadratic
discussed prominently.
But a very little material of solving standard quadratic
congruence of composite modulus is found. Only a method
known as Chinese Remainder Theorem (CRT) can be used
[3]. The author’s formulations of solutions of standard
ongruence of composite modulus are found [4],
ANALYSIS & RESULTS
Consider the congruence under consideration:
‫ݔ‬ଶ
≡ ܾ ሺ݉‫݌ ݀݋‬௡
‫ݍ‬௠ሻ;
Its individual congruence are:
‫ݔ‬ଶ
≡ ܾ ሺ݉‫݌ ݀݋‬௡ሻ………………………….(I)
‫ݔ‬ଶ
≡ ܾ ሺ݉‫ݍ ݀݋‬௠ሻ …
Case-I: For ܾ ൌ ܽଶ
, the congruence (I) has exactly two
solutions and the congruence (II) also has exactly two
solution. Therefore, the congruence under consideration
has exactly four incongruent solutions [1].
It is generally known that the standard quadratic
congruence of the type:
‫ݔ‬ଶ
≡ ܽଶ
ሺ݉‫݉ ݀݋‬ሻ has at least two solutions
ܽ, ݉ െ ܽ ሺ݉‫݉ ݀݋‬ሻ ሾ2ሿ.
Therefore, the two obvious solutions of the said
congruence are given by:
‫ݔ‬ ≡ േܽ ≡ ܽ, ‫݌‬௠
‫ݍ‬௠
െ ܽ ሺ݉‫݀݋‬
For other two solutions, consider
ܽሺ݉‫݉ݍ݊݌ ݀݋‬ሻ.
Then ‫ݔ‬ଶ
≡ ሺ‫݌‬௡
݇ േ ܽሻଶ
International Journal of Trend in Scientific Research and Development (IJTSRD)
ISSN: 2456 – 6470
2020 Page 326
of Standard
f Composite Modulus–
f Two Powered Odd Primes
Head, Department of Mathematics,
Maharashtra, India
How to cite this paper: Prof B M Roy
129: Formulation of a Special Type
of Standard Quadratic Congruence of
Composite Modulus– a Product of Two
Powered Odd
Primes" Published
in International
Journal of Trend in
Scientific Research
and Development
(ijtsrd), ISSN: 2456-
6470, Volume-4 |
4, June 2020, pp.326-328, URL:
www.ijtsrd.com/papers/ijtsrd30964.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/
Consider the congruence under consideration:
ሻ; ‫, ݌‬ ‫.ݏ݁݉݅ݎ݌ ݀݀݋ ݁ݎܽ ݍ‬
Its individual congruence are:
ሻ………………………….(I)
ሻ … … … … … … … … ሺ‫ܫܫ‬ሻ
the congruence (I) has exactly two
solutions and the congruence (II) also has exactly two
solution. Therefore, the congruence under consideration
has exactly four incongruent solutions [1].
that the standard quadratic
has at least two solutions ‫ݔ‬ ≡ േܽ ≡
Therefore, the two obvious solutions of the said
ሺ݉‫݌ ݀݋‬௠
‫ݍ‬௠ሻ.
r two solutions, consider ‫ݔ‬ ≡ േሺ‫݌‬௡
݇ േ
IJTSRD30964

2. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
@ IJTSRD | Unique Paper ID – IJTSRD30964 | Volume – 4 | Issue – 4 | May-June 2020 Page 327
≡ ሺ‫݌‬௡
݇ሻଶ
± 2. ‫݌‬௡
݇. ܽ + ܽଶ
ሺ݉‫݌ ݀݋‬௡
‫ݍ‬௠
ሻ
≡ ܽଶ
+ ‫݌‬௡
݇ ሺ ‫݌‬௡
݇ ± 2ܽሻ ሺ݉‫݌ ݀݋‬௠
‫ݍ‬௠
ሻ
≡ ܽଶ
+ ‫݌‬௡
. ‫ݍ‬௠
‫,ݐ‬ ݂݅ ݇ሺ ‫݌‬௡
݇ ± 2ܽሻ = ‫ݍ‬௠
‫ .ݐ‬
≡ ܽଶ
ሺ݉‫݌ ݀݋‬௡
‫ݍ‬௠
ሻ
Therefore, the other two solutions are given by:
‫ݔ‬ ≡ ±ሺ‫݌‬௡
݇ ± ܽሻሺ݉‫݌ ݀݋‬௡
‫ݍ‬௠
ሻ,
݂݅ ݇ሺ ‫݌‬௡
݇ ± 2ܽሻ = ‫ݍ‬௠
‫ .ݐ‬
Sometimes the congruence may be of the type:
‫ݔ‬ଶ
≡ ܾ ሺ݉‫݌ ݀݋‬௡
‫ݍ‬௠ሻ. Then at first the solvability condition
must be tested to know if the congruence is solvable.
If the Legendre’s symbols are (
௕
௣
ሻ = ቀ
௕
௤
ቁ = 1, then it is
said that the congruence is solvable [1].
Then the congruence must be written as: ‫ݔ‬ଶ
≡ ܾ +
݇. ‫݌‬௡
‫ݍ‬௠
= ܽଶ
ሺ݉‫݌ ݀݋‬௡
‫ݍ‬௠ሻ for a certain value of k [2]. Then
the four solutions are given by as in above.
Case-II: For ܾ = ‫ݍ‬ଶ
, the congruence (I) has exactly two
solutions and the congruence (II) also has exactly q-
solution. Therefore, the congruence under consideration
has exactly 2q- solution. Therefore, the congruence
‫ݔ‬ଶ
≡ ‫ݍ‬ଶ
ሺ݉‫݌ ݀݋‬௡
‫ݍ‬௠ሻ has exactly 2q- solutions.
These are given by
‫ݔ‬ ≡ ‫݌‬௡
‫ݍ‬௠ିଵ
݇ ± ‫ ݍ‬ሺ݉‫݌ ݀݋‬௡
‫ݍ‬௠ሻ ‫ݐ݅ݓ‬ℎ ݇ = 0, 1, 2, … … . , ‫.1ݍ‬
Case-III: ‫ܾ ݎ݋ܨ‬ = ‫݌‬ଶ
, the congruence (I) has exactly p-
solutions and the congruence (II) also has exactly two
solution. Therefore, the congruence under consideration
has exactly 2p solution. Therefore, the congruence
‫ݔ‬ଶ
≡ ‫݌‬ଶ
ሺ݉‫݌ ݀݋‬௡
‫ݍ‬௠ሻ has exactly 2p- solutions.
These are given by ‫ݔ‬ ≡ ‫݌‬௡ିଵ
‫ݍ‬௠
݇ ± ‫ ݍ‬ሺ݉‫݌ ݀݋‬௡
‫ݍ‬௠ሻ ݂‫݇ ݎ݋‬ =
0, 1, 2, … … … … , ‫݌‬ − 1.
ILLUSTRATIONS
Example-1: Consider the congruence:
‫ݔ‬ଶ
≡ 49 ሺ݉‫5211 ݀݋‬ሻ.
It can be written as: ‫ݔ‬ଶ
≡ 7ଶ
ሺ݉‫5 ݀݋‬ଷ
. 3ଶሻ.
It is of the type: ‫ݔ‬ଶ
≡ ܽଶ
ሺ݉‫݌ ݀݋‬௡
‫ݍ‬௠ሻ ‫ݐ݅ݓ‬ℎ ‫݌‬ = 5, ‫ݍ‬ = 3.
It has four Solutions.
Two of the solutions are given by
‫ݔ‬ ≡ ±ܽ ≡ ܽ, ‫݌‬௡
‫ݍ‬௠
− ܽ ሺ݉‫݌ ݀݋‬௡
‫ݍ‬௠ሻ.
≡ ±7 ≡ 7, 1125 − 3 ሺ݉‫5211 ݀݋‬ሻ
≡ 7, 1118 ሺ݉‫5211 ݀݋‬ሻ.
Other two solutions are given by
‫ݔ‬ ≡ ±ሺ‫݌‬௡
݇ ± ܽሻ, ݂݅ ݇ሺ‫݌‬௡
݇ ± 2ܽሻ = ‫ݍ‬௠
. ‫ݐ‬
≡ ±ሺ125݇ ± 7ሻ, ݂݅ ݇ሺ125݇ ± 2.7ሻ = 3ଶ
‫ݐ‬
≡ ±125݇ ± 7ሻ, ݂݅ ݇ሺ125݇ ± 14ሻ = 9‫.ݐ‬
But for ݇ = 4, ‫ݔ‬ ≡ ±ሺ125.4 − 7ሻ ܽ‫.4 ݏ‬ ሺ125.4 − 14ሻ =
4. ሺ500 − 14ሻ = 216.9
≡ ±493 ሺ݉‫5211 ݀݋‬ሻ
≡ 493, 632 ሺ݉‫5211 ݀݋‬ሻ.
Therefore, ‫ݔ‬ ≡ 7, 1118; 493, 632 ሺ݉‫5211 ݀݋‬ሻ.
Example -2: Consider the congruence:‫ݔ‬ଶ
≡ 31 ሺ݉‫522 ݀݋‬ሻ.
It can be written as:
‫ݔ‬ଶ
≡ 31 + 225 = 256 = 16ଶ
ሺ݉‫5 ݀݋‬ଶ
. 3ଶሻ.
It is of the type: ‫ݔ‬ଶ
≡ ܽଶ
ሺ݉‫݌ ݀݋‬௡
‫ݍ‬௠ሻ ‫ݐ݅ݓ‬ℎ ‫݌‬ = 5, ‫ݍ‬ = 3.
It has four Solutions.
Two of the solutions are given by
‫ݔ‬ ≡ ±ܽ ≡ ܽ, ‫݌‬௡
‫ݍ‬௠
− ܽ ሺ݉‫݌ ݀݋‬௡
‫ݍ‬௠ሻ.
≡ ±16 ≡ 16, 225 − 16 ሺ݉‫5211 ݀݋‬ሻ
≡ 16, 209 ሺ݉‫522 ݀݋‬ሻ.
Other two solutions are given by
‫ݔ‬ ≡ ±ሺ‫݌‬௡
݇ ± ܽሻ, ݂݅ ݇ሺ‫݌‬௡
݇ ± 2ܽሻ = ‫ݍ‬௠
. ‫ݐ‬
≡ ±ሺ25݇ ± 16ሻ, ݂݅ ݇ሺ25݇ ± 2.16ሻ = 3ଶ
‫ݐ‬
≡ ±ሺ25݇ ± 16ሻ, ݂݅ ݇ሺ25݇ ± 32ሻ = 9‫.ݐ‬
But for ݇ = 2, ‫ݔ‬ ≡ ±ሺ25.2 − 16ሻ ܽ‫.2 ݏ‬ ሺ25.2 − 32ሻ =
2. ሺ50 − 32ሻ = 2.18 = 36 = 9.4
≡ ±ሺ25.2 − 16ሻ ሺ݉‫522 ݀݋‬ሻ
≡ 34, 191 ሺ݉‫522 ݀݋‬ሻ.
Therefore, ‫ݔ‬ ≡ 16, 209; 34, 191 ሺ݉‫522 ݀݋‬ሻ.
Example-3: Consider the congruence:
‫ݔ‬ଶ
≡ 25 ሺ݉‫5216 ݀݋‬ሻ.
It can be written as: ‫ݔ‬ଶ
≡ 5ଶ
ሺ݉‫7 ݀݋‬ଶ
. 5ଷሻ.
It is of the type: ‫ݔ‬ଶ
≡ ‫ݍ‬ଶ
ሺ݉‫݌ ݀݋‬௡
‫ݍ‬௠ሻ ‫ݐ݅ݓ‬ℎ ‫݌‬ = 7, ‫ݍ‬ = 5.
It has 2.5=10 solutions.
The solutions are given by
‫ݔ‬ ≡ ‫݌‬௡
‫ݍ‬௠ିଵ
݇ ± ‫ ݍ‬ሺ݉‫݌ ݀݋‬௡
‫ݍ‬௠ሻ; ݇ = 0, 1, 2,3, 4.
≡ 7ଶ
. 5ଶ
݇ ± 5 ሺ݉‫7 ݀݋‬ଶ
. 5ଷ
ሻ
≡ 1225݇ ± 5 ሺ݉‫5216 ݀݋‬ሻ.
5, 6120; 1220, 1230; 2445, 2455; 3670, 3680; 4895,4905 ሺ݉‫5216 ݀݋‬ሻ
Example-4: Consider the congruence:
‫ݔ‬ଶ
≡ 121 ሺ݉‫30514 ݀݋‬ሻ.
It can be written as: ‫ݔ‬ଶ
≡ 11ଶ
ሺ݉‫343.121 ݀݋‬ሻ i.e.
‫ݔ‬ଶ
≡ 11ଶ
ሺ݉‫11 ݀݋‬ଶ
. 7ଷ
ሻ.
It is of the type: ‫ݔ‬ଶ
≡ ‫݌‬ଶ
ሺ݉‫݌ ݀݋‬௡
‫ݍ‬௠ሻ ‫ݐ݅ݓ‬ℎ ‫݌‬ = 11, ‫ݍ‬ = 7.
It has twenty two solutions.
The solutions are given by
‫ݔ‬ ≡ ‫݌‬௡ିଵ
‫ݍ‬௠
݇ ± ‫ ݌‬ሺ݉‫݌ ݀݋‬௡
‫ݍ‬௠ሻ; ݇ = 0, 1, 2, … … … ,10.
≡ 11. 7ଷ
݇ ± 11 ሺ݉‫11 ݀݋‬ଶ
. 7ଷሻ.
≡ 3773݇ ± 11 ሺ݉‫30514 ݀݋‬ሻ
≡ ±11; 3773 ± 11; 7546 ± 11; 11319 ± 11; 15092 ±
11; 18865 ± 11;
22638 ± 11; 26411 ± 11; 30184 ± 11; 33957 ±
11; 37730 ± 11 ሺ݉‫77 ݀݋‬ሻ.
≡ 11, 41492; 3762, 3784; 7535, 7557;
11308, 11330; … … . . ; 37719, 37741 ሺ݉‫30514 ݀݋‬ሻ.
Example-5: Consider the congruence:
‫ݔ‬ଶ
≡ 625 ሺ݉‫5202 ݀݋‬ሻ.
It can be written as: ‫ݔ‬ଶ
≡ 625 ሺ݉‫18.52 ݀݋‬ሻ ݅. ݁. ‫ݔ‬ଶ
≡
ሺ25ሻଶ
ሺ݉‫5 ݀݋‬ଶ
. 3ସሻ.
It is of the type: ‫ݔ‬ଶ
≡ ሺ݉‫݌‬ሻଶ
ቀ݉‫݌ ݀݋‬௡
‫ݍ‬௠ቁ ‫ݐ݅ݓ‬ℎ ‫݌‬ = 5, ‫ݍ‬ =
3.
It has exactly ten solutions given by
‫ݔ‬ ≡ ‫݌‬௡ିଵ
‫ݍ‬௠
݇ ± ݉‫ ݌‬ሺ݉‫݌ ݀݋‬௡
‫ݍ‬௠ሻ; ݇ = 0, 1, 2, 3, 4.
≡ 5ଵ
. 3ସ
݇ ± 5.5 ሺ݉‫5 ݀݋‬ଶ
. 3ସ
ሻ
≡ 405݇ ± 25 ሺ݉‫5202 ݀݋‬ሻ; ݇ = 0, 1, 2, 3, 4.
≡ 0 ± 25; 405 ± 25; 810 ± 25; 1215 ± 25; 1620 ±
25 ሺ݉‫5202 ݀݋‬ሻ
≡25, 2000; 380,430; 785, 835; 1190, 1240; 1595, 1645
ሺ݉‫5202 ݀݋‬ሻ

3. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
@ IJTSRD | Unique Paper ID – IJTSRD30964 | Volume – 4 | Issue – 4 | May-June 2020 Page 328
CONCLUSION
Therefore, it is concluded that the congruence:
‫ݔ‬ଶ
≡ ܽଶ
ሺ݉‫݌ ݀݋‬௡
. ‫ݍ‬௠ሻ has exactly four solutions, two of
them are given by ‫ݔ‬ ≡ ±ܽ ≡ ܽ, ‫݌‬௡
. ‫ݍ‬௠
− ܽ ሺ݉‫݌ ݀݋‬௡
. ‫ݍ‬௠ሻ.
Remaining two are given by
‫ݔ‬ ≡ ±ሺ‫݌‬௡
݇ ± ‫݌‬ሻ ሺ݉‫݌ ݀݋‬௡
. ‫ݍ‬௠ሻ, ݂݅ ݇ሺ‫݌‬௡
݇ ± 2ܽሻ = ‫ݍ‬௠
. ‫ݐ‬
It is also concluded that the congruence
‫ݔ‬ଶ
≡ ‫݌‬ଶ
ሺ݉‫݌ ݀݋‬௡
. ‫ݍ‬௠ሻ has exactly 2p-solutions given by
‫ݔ‬ ≡ ‫݌‬௡ିଵ
. ‫ݍ‬௠
݇ ± ‫;݌‬ ݇ = 0, 1, 2, … … . , ‫݌‬ − 1.
It is also concluded that the congruence
‫ݔ‬ଶ
≡ ‫ݍ‬ଶ
ሺ݉‫݌ ݀݋‬௡
. ‫ݍ‬௠ሻ has exactly 2q-solutions given by
‫ݔ‬ ≡ ‫݌‬௡
. ‫ݍ‬௠ିଵ
݇ ± ‫;ݍ‬ ݇ = 0, 1, 2, … … . , ‫ݍ‬ − 1.
REFERENCE
[1] Niven I., Zuckerman H. S., Montgomery H. L. (1960,
Reprint 2008), “An Introduction to The Theory of
Numbers”, 5/e, Wiley India ( Pvt ) Ltd.
[2] Roy B M, “Discrete Mathematics & Number Theory”,
1/e, Jan. 2016, Das Ganu Prakashan, Nagpur.
[3] Thomas Koshy, “Elementary Number Theory with
Applications”, 2/e (Indian print, 2009), Academic
Press.
[4] Roy B M, Formulation of solutions of Solvable
standard quadratic congruence of odd composite
modulus-a product of two odd primes and also a
product of twin primes, International Journal of
Current Research (IJCR), ISSN: 0975-833X, Vol-10,
Issue-05, May-18.
[5] Roy B M, Formulation of solutions of a class of solvable
standard quadratic congruence of composite modulus-
an odd prime positive integer multiple of five,
International Journal for Research Trends and
Innovations (IJRTI), ISSN: 2456-3315, Vol-03, Issue-
9, Sep-18.
[6] Roy B M, Formulation of solutions of a class of solvable
standard quadratic congruence of composite modulus-
an odd prime positive integer multiple of seven,
International Journal of Science & Engineering
Development Research (IJSDR), ISSN: 2455-2631,
Vol-03, Issue-11, Nov-18.
[7] Roy B M, Formulation of solutions of a very special
class of standard quadratic congruence of multiple of
prime-power modulus, International Journal of
Scientific Research and Engineering Development
(IJSRED), ISSN: 2581-7175, Vol-02, Issue-06, Nov-
Dec-19
[8] Roy B M, Formulation of solutions of a standard
quadratic congruence of composite modulus-an odd
prime multiple of power of an odd prime,
International Journal of Scientific Research and
Engineering Development (IJSRED), ISSN: 2581-
7175, Vol-03, Issue-02, Mar-Apr-20.