Loading ...
IJTSRD
Education
Volume 4 Issue 5
10
0
Try Now
Log In
Pricing
International Journal of Trend in Scientific Research and Development Volume 5 Issue 4, May-June @ IJTSRD | Unique Paper ID – IJTSRD42321 RP-166: Solving Some Special Standard Cubic Congruence Composite Modulus Head, Department of Mathematics, Jagat Arts, Commerce & ABSTRACT Here in this paper, ten special type of standard cubic congruence of composite modulus are studied for their solutions. It is found that each of the cubic congruence under consideration has a single solution. The solution can be obtained orally as the solu extra effort is necessary to find the solution. KEYWORDS: Cubic Congruence, Composite Modulus, Unique Solution INTRODUCTION Some standard cubic congruence of special type are considered for study and are formulated their solutions. All the considered cubic congruence have unique solutions. Those solutions are present in the congruence itself. Here is the list of those cubic Congruence in the problem statement. PROBLEM-STATEMENT “To find formula for solutions of the congruence: ≡ 2 , ≡ 3 , ≡ 2 3 , ≡ 4 , ≡ 3 4 , ≡ 6 , ≡ 2 6 , ≡ 3 6 , ≡ 4 6 , ≡ 5 6 , . LITERATURE REVIEW The standard cubic congruence found no place in the literature of mathematics as it is not studied; it is not a part of syllabus in the university course. Only linear 2021 Available Online: www.ijtsrd.com e-ISSN: 2456 | Volume – 5 | Issue – 4 | May-June 202 modulo a Multiple of an Odd Prime Prof B M Roy I H P Science College, Goregaon tion is given in the problems. No How to cite this paper "RP-166: Solving Some Special Standard Cubic Congruence of Composite Modulus modulo a Multiple of an Odd Prime" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456 6470, Volume Issue-4, June 2021, pp.551-553, www.ijtsrd.com/papers/ijtsrd42321.pdf Copyright © 20 International Journal Scientific Research and Development Journal. This is an Open Access article distributed un the terms Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0 " congruence of degree one and standard quadratic congruence of prime and composite modul remained in the part of study [1], [2], [3].Also some of the author’s papers are seen [4], [5], [6]. ANALYSIS & RESULTS Consider the congruence: odd prime. It is easily seen that: 0 8 as . Hence ≡ 2 is a solution of the congruence. Consider the congruence: odd prime. It is easily seen that: 0 3 as . Hence ≡ 3 is a solution of the congruence. Consider the congruence: odd prime. It is easily seen that: 2 . 3 ≡ 0 3 as Hence ≡ 2 3 is a solution of the congruence. Consider the congruence: odd prime. It is easily seen that: 1 . 4 ≡ 0 4 as (IJTSRD) – 6470 1 Page 551 of , Maharashtra, India : Prof B M Roy - -5 | URL: 21 by author (s) and of Trend in der of the (CC BY 4.0) ) us are ≡ 2 . Here p is an ! 1 . 2 ≡ ≡ 3 . Here p is an 1 " # 1 . 3 ≡ ≡ 2 3 . Here p is an 2 22 1 2" # 1 . ≡ 4 . Here p is an ! 1 1 # . IJTSRD42321 International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD42321 | Volume – 5 | Issue – 4 | May-June 2021 Page 552 Hence ≡ 4 is a solution of the congruence. Consider the congruence: ≡ 3 4 . Here p is an odd prime. It is easily seen that: 3 3 39! 1 3[4! # 4! # ! 1 ] 3. 4 ≡ 0 4 . Hence ≡ 3 4 is a solution of the congruence. Consider the congruence: ≡ 8 . Here p is an odd prime. It is easily seen that: ! 1 . 8 ≡ 0 8 as ! ≡ 1 8 . Hence ≡ 8 is a solution of the congruence Consider the congruence: ≡ 3 8 . Here p is an odd prime. It is easily seen that: 3 3 39! 1 3[8! # ! 1 ] 3. 8 ≡ 0 8 . Hence ≡ 8 is a solution of the congruence. Consider the congruence: ≡ 5 8 . Here p is an odd prime. It is easily seen that: 5 5 525! 1 5[24! # ! 1 ] 5. 8 ≡ 0 8 . Hence ≡ 5 8 is a solution of the congruence. Consider the congruence: ≡ 7 8 . Here p is an odd prime. It is easily seen that: 7 7 749! 1 7[48! # ! 1 ] 7. 8 ≡ 0 8 . Hence ≡ 7 8 is a solution of the congruence. ILLUSTRATIONS Example-1:Consider the congruence ≡ 7 14 . It can be written as ≡ 7 2.7 . It is of the type ≡ 2 ( ℎ 7. It has single solution ≡ 2 ≡ 7 2.7 ≡ 7 14 . Consider the congruence ≡ 7 21 . It can be written as ≡ 7 3.7 . It is of the type ≡ 3 ( ℎ 7. It has single solution ≡ 3 ≡ 7 3.7 ≡ 7 21 . Consider the congruence ≡ 14 21 . It can be written as ≡ 2.7 3.7 . It is of the type ≡ 2 3 ( ℎ 7. It has single solution ≡ 2 3 ≡ 2.7 3.7 ≡ 14 21 . Example-1: Consider the congruence ≡ 7 56 . It can be written as ≡ 7 8.7 . It is of the type ≡ 8 ( ℎ 7. It has single solution ≡ 8 ≡ 7 8.7 ≡ 7 56 . Example-2: Consider the congruence ≡ 21 56 . It can be written as ≡ 3.7 8.7 . It is of the type ≡ 3 8 ( ℎ 7. It has single solution ≡ 3 8 ≡ 3.7 8.7 ≡ 21 56 . Example-3: Consider the congruence ≡ 35 56 . It can be written as ≡ 5.7 8.7 . It is of the type ≡ 5 8 ( ℎ 7. It has single solution ≡ 5 8 ≡ 5.7 8.7 ≡ 35 56 . Example-4: Consider the congruence ≡ 49 56 . It can be written as ≡ 7.7 8.7 . It is of the type ≡ 7 8 ( ℎ 7. It has single solution ≡ 7 8 ≡ 7.7 8.7 ≡ 49 56 . CONCLUSION It can be concluded from this discussion that the standard cubic congruence considered, each has a single solutions. It is found that the congruence ≡ 2 , p an odd prime has a unique solution ≡ 2 . The congruence ≡ 3 , p an odd prime has a unique solution ≡ 3 . The congruence ≡ 2 3 , p an odd prime has a unique solution ≡ 2 3 . The congruence ≡ 4 , p an odd prime has a unique solution ≡ 4 . The congruence ≡ 3 4 , p an odd prime has a unique solution ≡ 3 4 . The congruence ≡ 8 , p an odd prime has a unique solution International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD42321 | Volume – 5 | Issue – 4 | May-June 2021 Page 553 ≡ 8 . The congruence ≡ 3 8 , p an odd prime has a unique solution ≡ 3 8 . The congruence ≡ 2 3 , p an odd prime has a unique solution ≡ 2 3 . The congruence ≡ 5 8 , p an odd prime has a unique solution ≡ 5 8 . The congruence ≡ 7 8 , p an odd prime has a unique solution ≡ 7 8 . MERIT OF THE PAPER The use of Chinese remainder theorem is needless. Solutions can be obtained orally. This is the merit of the paper. REFERENCE [1] Zuckerman H. S., Niven I., 2008, An Introduction to the Theory of Numbers, Wiley India, Fifth Indian edition, ISBN: 978-81-265-1811-1. [2] David M Burton, 2012, Elementary Number Theory, McGraw Hill education (Higher Education), Seventh Indian Edition, New Dehli, India, ISBN: 978-1-25- 902576-1. [3] Thomas Koshy, 2009, Elementary Number Theory with Applications, Academic Press, Second Edition, Indian print, New Dehli, India, ISBN: 978-81-312- 1859-4 [4] Roy B M, Formulation of a class of standard cubic congruence modulo a positive prime integer multiple of nine, ISSN: International Journal of Recent Innovations in Academic Research (IJRIAR), ISSN: 2635-3040, vol-02, Issue-05, Sept-18. [5] Roy B M, Formulation of solutions of a class of standard cubic congruence modulo *+power of an integer multiple of *+ power of three, International Journal of Recent Innovations in Academic Research (IJRIAR), ISSN: 2635-3040, Vol-03, Issue- 01, Jan-19. [6] Roy B M, Formulation of Two Special Classes of Standard Cubic Congruence of Composite Modulus—a power of three, International Journal of Scientific Research and Engineering Development (IJSRED), 2581-7175,Vol-02, Issue-03,May-19. [7] Roy B M, Solving some special standard cubic congruence modulo an odd prime multiplied by eight, International Journal of Scientific Research and Engineering Development(IJSRED), ISSN: 2581- 7175, Vol-04, Issue-01, Jan-21.
