10 VIII August 2022 https://doi.org/10.22214/ijraset.2022.46147
(3) Below, we present
A
found by 223
of distinct
patterns (1) A. Pattern: 1 Assume
Zzabab (4)
International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321 9653; IC Value: 45.98; SJ Impact Factor: 7.538 Volume 10 Issue VIII Aug 2022 Available at www.ijraset.com 231©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 | Properties of the Ternary Cubic Equation 22353xyz G. Janaki1, A. Gowri Shankari2 1Associate Professor, 2Assistant Professor, PG & Research Department of Mathematics, Cauvery College for Women (Autonomous), Affiliated to Bharathidasan University, Trichy 18
A. Notations Tm, a: Polygonal Number of rank a with side m Gnoa: Gnomonic Number of rank a Stara: Star Number of rank a Oa: Octahedral Number of rank a Pa m: Pyramidal Number of rank a with sides m SOa: Stella Octangula Number of rank a CCa: Centered Cube Number of rank a CSa: Centered Square Number of rank a RDa: Rhombic Dodecagonal Number of rank a TOa: Truncated Octahedral Number of rank a II.
the
Abstract: To identify its different integral non zero solutions, the ternary cubic equation 22353xyz is taken into consideration. Different integral solution patterns to the ternary cubic equation under consideration are obtained in each pattern by using the linear transformation and the method of factorization; interesting relationships between the solutions and some polygonal numbers, such as pyramidal and central pyramidal numbers, are also displayed. Keywords: Diophantine equations, Ternary equation, Cubic Equation with Three Unknowns, Integral Solutions I. INTRODUCTION Number theory, which is used to explain anything that can be quantified, is the language of patterns and relationships. A polynomial equation called a Diophantine equation can only have integers as solutions. Theories of numbers were featured in [1 3]. A unique Pythagorean triangle problem and its integral solutions are featured in [4, 5]. Higher order equations are taken into account for integral solutions in [6 10]. The non homogeneous cubic equation with three unknowns represented by the equation is discussed in this communication, and in particular, a few intriguing relationships between the solutions are highlighted. METHOD OF ANALYSIS non zero integral solution to Cubic equation can be 53xyz 223 154 XTZ illustration integer zero 22(,)15
non
(1) Upon switching the transformations, xXTyXTzZ 3,5,2 (2) in (1) leads to,
International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321 9653; IC Value: 45.98; SJ Impact Factor: 7.538 Volume 10 Issue VIII Aug 2022 Available at www.ijraset.com 232©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 | Where a and b are positive integers. And write 4(8215)(8215) (5) Using the factorization approach, replacing (3) with (4) and (5), 33 (15)(15)(8215)(8215)(15)(15) XTXTabab Comparing real and imaginary elements while equating similar phrases, 3223 836090450 Xaababb 3223 29024120 Taababb The appropriate integer solutions of equation (1) are provided by substituting the above mentioned values of X and T into equation (2) 22 3223 3223 (,)230 (,)188102101050 (,)14630162810 zzabab yyabaababb xxabaababb Properties: 1. 7(,) zaa and 28(,) zaa are Perfect Squares 2. (1,1)(1,1)(1,1) xyz is a Harshad Number 3. (,1)(,1)62239(mod123) 29,26, 6 aaaayaxaPstarTT 4 4051)0(mod912,1)(124,1)(,1)( aa yaxazaSOGno 5. (,1)(,1)1841076(mod852) 3 aa yazaPStar 6. (,1)(,1)16(,1)2292352(mod1769) 30, aaxayazaTOT 7 (,1)(,1)23(,1)12930(mod187) 18, 5 aaxayazaPT B. Pattern: 2 The modified form of equation (3) is 154*1223 XTZ (6) Put 4 in as, 4(8215)(8215) (7) and 1(415)(415) (8) When (7) and (8) are substituted in equation (6) and the process of factorization is used, as described in Pattern 1, the corresponding integer solutions of (1) are represented by 22 3223 3223 (,)230 (,)142639016508250 (,)110495012786390 zzabab yyabaababb xxabaababb Properties: 1. 3,4(1,1)(1,1)(1,1)22 yxzT is a nasty number 2. 3,4(1,1)(1,1)26 xyT is a perfect square number 3. 8,3(1,1)(1,1)11(1,1) yxzT Is a cubic number 4 (,1)(,1)8728860(mod2762) 15, aaa yaxaRDTGno 5. 0(mod8152)71248,1)(186,1)(,1)( aa yaxazaOGno
[1] Carmichael R.D., “The Theory of Numbers and Diophantine Analysis”, Dover Publications, New York, 1959 [2] Dickson L.E., “History of the theory of numbers”, Chelsia Publishing Co., Vol II, New York, 1952 [3] Mordell L.J., “Diophantine Equations”, Academic Press, London 1969 [4] Telang S. G., “Number Theory”, Tata Mc Graw Hill Publishing Company, New Delhi 1996
International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321 9653; IC Value: 45.98; SJ Impact Factor: 7.538 Volume 10 Issue VIII Aug 2022 Available at www.ijraset.com 233©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 | 6. (,1)(,1)2471392(mod1815) 22, aayaxaTOT 7. 33000(mod6319)213,1)(825,1)( ayaxaO 8. 2(,1)426(,1)11028465670(mod6567) 26, aaa xazaSOTGno C. Pattern: 3 Write 1(31815)(31815) (9) By replacing (7) and (9) in equation (6) and factorising the result using the steps in Pattern 1, the corresponding integer solutions of (1) are represented by 22 3223 3223 (,)230 (,)1118503101299064950 (,)866389701006250310 zzabab yyabaababb xxabaababb Properties: 1. (,1)16(,1)22361184304830(mod95913) 22, 5 aaa yazaPTGno 2. (,1)(,1)(,1)18982757960(mod23056) 10, aaa yaxazaHOTGno 3. 10((1,1)(1,1)) 1 xy is a square number 4. (1,)(1,)23(1,)1383123239442950(mod42086) 7 bbb xbybzbPStarGno 5. 3,4(1,1)(1,1)24 yxT is a nasty number 6. (,1)16(,1)4336113292265839894(mod71658) 3, aaa xazaCCCST 7. (,1)(,1)23(,1)6323914013(mod14195) 30, aaxayazaRDT 8. (,1)(,1)19(,1)126148314070(mod12949) 6, aayaxazaSOT
Note: Additionally, 4 and 1 might be written as 49 15)15)(8(81 151615)(6216(624 In relation to these options, one may find many patterns of solutions of (1) III. CONCLUSION Three unique patterns of non zero distinct integer solutions to the given non homogeneous problem 22353xyz are shown in this paper. For various options of cubic Diophantine equations, additional patterns of non zero integer unique solutions and their corresponding characteristics may be found. REFERENCES
International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321 9653; IC Value: 45.98; SJ Impact Factor: 7.538 Volume 10 Issue VIII Aug 2022 Available at www.ijraset.com 234©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 | [5] Gopalan M.A, Manju Somnath and Vanitha N, Parametric Solutions of 262 xyz , Acta Ciencia Indica, Vol XXXIII, 3, 1083 1085, 2007 [6] Gopalan M.A and Janaki G, “Integral solutions of ()(332)2()" 2222223 xyxyxyzwp , Impact J. Sci, Tech., 4(1), 97 102, 2010 [7] Gopalan M.A, Pandichelvi V, “Observations on the ternary cubic equation 4()332232 xyxyzz ”, Archimedes J. Math 1(1), 31 37, 2011 [8] Gopalan M.A and Janaki G, “Integral solutions of 2222223 ()(332)2() xyxyxyzwp ”, Impact Journal of Science & Technology, Vol 4, No. 97 102, 2010 [9] Janaki G, Saranya C, “Observations on Ternary Quadratic Diophantine Equation 6()1133972" 222 xyxyxyz , International Journal of Innovative Research and Science, Engineering and Technology, Volume 5, Issue 2, February 2016 [10] Janaki G, Saranya P, “ On the Ternary Cubic Diophantine Equation "404)4(6)5( 223 xyxyxyz , International Journal of Science and Research online, Vol 5, Issue 3, Pg. No: 227 229, March 2016
