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 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. Prof B M Roy "RP-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 | Issue-4 , June 2020, URL: https://www.ijtsrd.com/papers/ijtsrd30964.pdf Paper Url :https://www.ijtsrd.com/mathemetics/applied-mathematics/30964/rp129-formulation-of-a-special-type-of-standard-quadratic-congruence-of-composite-modulus%E2%80%93-a-product-of-two-powered-odd-primes/prof-b-m-roy
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 ݀
ݍሻ. 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 ݀
ݍሻ 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 ݀
ݍሻ has exactly 2q- solutions.
These are given by
ݔ ≡
ݍିଵ
݇ ± ݍሺ݉ ݀
ݍሻ ݐ݅ݓℎ ݇ = 0, 1, 2, … … . , .1ݍ
Case-III: ܾ ݎܨ = 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 ݀
ݍሻ has exactly 2p- solutions.
These are given by ݔ ≡ ିଵ
ݍ
݇ ± ݍሺ݉ ݀
ݍሻ ݂݇ 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 ݀
ݍሻ ݐ݅ݓℎ = 5, ݍ = 3.
It has four Solutions.
Two of the solutions are given by
ݔ ≡ ±ܽ ≡ ܽ,
ݍ
− ܽ ሺ݉ ݀
ݍሻ.
≡ ±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 ݀
ݍሻ ݐ݅ݓℎ = 5, ݍ = 3.
It has four Solutions.
Two of the solutions are given by
ݔ ≡ ±ܽ ≡ ܽ,
ݍ
− ܽ ሺ݉ ݀
ݍሻ.
≡ ±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 ݀
ݍሻ ݐ݅ݓℎ = 7, ݍ = 5.
It has 2.5=10 solutions.
The solutions are given by
ݔ ≡
ݍିଵ
݇ ± ݍሺ݉ ݀
ݍሻ; ݇ = 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 ݀
ݍሻ ݐ݅ݓℎ = 11, ݍ = 7.
It has twenty two solutions.
The solutions are given by
ݔ ≡ ିଵ
ݍ
݇ ± ሺ݉ ݀
ݍሻ; ݇ = 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
≡ ሺ݉ሻଶ
ቀ݉ ݀
ݍቁ ݐ݅ݓℎ = 5, ݍ =
3.
It has exactly ten solutions given by
ݔ ≡ ିଵ
ݍ
݇ ± ݉ ሺ݉ ݀
ݍሻ; ݇ = 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 ݀
. ݍሻ has exactly four solutions, two of
them are given by ݔ ≡ ±ܽ ≡ ܽ,
. ݍ
− ܽ ሺ݉ ݀
. ݍሻ.
Remaining two are given by
ݔ ≡ ±ሺ
݇ ± ሻ ሺ݉ ݀
. ݍሻ, ݂݅ ݇ሺ
݇ ± 2ܽሻ = ݍ
. ݐ
It is also concluded that the congruence
ݔଶ
≡ ଶ
ሺ݉ ݀
. ݍሻ has exactly 2p-solutions given by
ݔ ≡ ିଵ
. ݍ
݇ ± ; ݇ = 0, 1, 2, … … . , − 1.
It is also concluded that the congruence
ݔଶ
≡ ݍଶ
ሺ݉ ݀
. ݍሻ has exactly 2q-solutions given by
ݔ ≡
. ݍିଵ
݇ ± ;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.