IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) DR. N.THIRUNIRAISELVI1* https://ijojournals.com/ Volume 07 || Issue 05|| May, 2024 || Techniques to solve Diophantine Equation of Degree Ten with Six Unknowns 822366 R)qp(800z3456yx  DR. N.THIRUNIRAISELVI1*, DR. M.A.GOPALAN2 1 Assistant Professor, Department of Mathematics, School of Engineering and Technology, Dhanalakshmi Srinivasan University, Samayapuram, Trichy- 621 112, Tamil Nadu, India. 2 Professor, Department of Mathematics, Shrimati Indira Gandhi College, Affiliated to Bharathidasan University, Trichy-620 002,Tamil Nadu, India. Abstract: This paper focuses on finding varieties of distinct integer solutions to the Diophantine equation of degree ten with six unknowns given by 822366 R)qp(800z3456yx  through employing the substitution strategy and method of factorization. A new representation for the factorization of integer 40 involving sides of Pythagorean triangle has been introduced. Key words: Higher degree Diophantine equation, Integer solution. 2020 Mathematics Subject Classification: 11D41. Introduction: It is well-known that the subject of Diophantine equations occupies a pivotal role in the Number theory. There is a vast general theory for higher degree Diophantine equations in many variables and it is a topic for research even today. While collecting problems on Diophantine equations of degree ten with six unknowns, the following paper [1] has been noticed and the authors have presented only few sets of integer solutions. It is worth to mention that the equation presented in [1] has many more fascinating patterns of solutions in integers. In this paper, the process of obtaining some more choices of solutions in integers to the Diophantine equation in title is illustrated. Substitution strategy and factorization method are applied successfully to obtain different choices of integer solutions to the considered equation. It is to be noted that a new representation for the factorization of integer 40 involving sides of Pythagorean triangle has been introduced. IJO JOURNALS Volume 07 | Issue 05 | May 2024 | https://ijojournals.com/index.php/m/index 1 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) DR. N.THIRUNIRAISELVI1* https://ijojournals.com/ Volume 07 || Issue 05|| May, 2024 || Method of Analysis: The Diophantine Equation is 822366 R)qp(800z3456yx  (1) To start with, it is observed by inspection that (1) is satisfied by the following sextuples {x, y, z, p, q, R} = {8u4v, 4u4v,2u8v, 18u4v,-6u4v,u2v}, {8*402*(9a2+4b2)2, 4*402*(9a2+4b2) 2, 2*404*(9a2+4b2)4, 18*402*(9a2+4b2)2, -6*402*(9a2+4b2)2, 40*(9a2+4b2)}, {32w2,16w2,32w2,72w2,24w2,2w}. However there are often fascinating solution patterns to (1) that are illustrated as follows. Introduction of the transformation s6t6q,s6t6p,stz,t2s3y,t2s3x  (2) in (1) leads to 422 R40t4s9  (3) Solving (3) through different ways and using (2), one obtains many non-zero distinct integer solutions to (1). Pattern 1: The choice Rt  (4) in leads to         9 1R10 R4s 2 22 (5) Let 19R10 22   (6) The smallest positive integers to (6) is 1R,1 00  Let 0101 h,RhR   (7) be the second solution to (6). Substituting (7) and (6) and performing some algebra, we get IJO JOURNALS Volume 07 | Issue 05 | May 2024 | https://ijojournals.com/index.php/m/index 2 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) DR. N.THIRUNIRAISELVI1* https://ijojournals.com/ Volume 07 || Issue 05|| May, 2024 || 00 18R20h  In view of (7), we get 001001 19R20,18R19R   which is written in matrix form as t 00 t 11 )R(M),R(   where        1920 1819 M and t is the transpose. The repetition of the above process leads to the general solution ),R( nn  to (6) as t 00 nt nn ),R(M),R(   Let  ~ ,~ be the eigen values of the matrix M. Then, it is found that 10619 ~ ,10619~   It is well known that )I~M(~~ ~ )I ~ M(~~ ~ M nn n            where I is the unit matrix of order 2. Thus, t 00nnnn nnnn t nn ),R( 2 ~~ 103 ) ~~(5 102 ) ~~(3 2 ~~ ),R(                    From (5), we have t 00 nt nn ),R(M),R(   From (5), we have nnn R2s  In view of (2), the corresponding integer solutions to (1) are given by IJO JOURNALS Volume 07 | Issue 05 | May 2024 | https://ijojournals.com/index.php/m/index 3 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) DR. N.THIRUNIRAISELVI1* https://ijojournals.com/ Volume 07 || Issue 05|| May, 2024 || nn nnn nnn nn 2 n nnn nnn Rt )21(R6q )21(R6p R2z )13(R2y )13(R2x            where . 2 ~~ 103 ) ~~(5 , 102 ) ~~(3 2 ~~ R nnnn n nnnn n            Pattern 2: Taking kR2s  (8) in (3), it is written as )k9R10(Rt 2222  (9) Let )k9R10( 222  (10) which is satisfied by k,kR 00   To obtain the other solutions to (10), consider the corresponding pell equation 1R10 22  whose general solution )~,R ~ ( nn  is given by nn nn f 2 1~ ,g 102 1 R ~    (11) Applying the lemma of Brahmagupta between ),R( 00  and )~,R ~ ( nn  , the general solution ),R( 1n1n   to (10) is given by IJO JOURNALS Volume 07 | Issue 05 | May 2024 | https://ijojournals.com/index.php/m/index 4 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) DR. N.THIRUNIRAISELVI1* https://ijojournals.com/ Volume 07 || Issue 05|| May, 2024 || ,g 102 k f 2 k R nn1n  (12) .g 10 k5 f 2 k nn1n   From (8) and (9), we get 1n1n1n 1n1n Rt Rk2s      In view of (2), the corresponding integer solutions to (1) are given by, )k2)(g 102 k f 2 k (3q ),k2)(g 102 k f 2 k (3p ),g10f10()gf10( 1040 k z ),2k6)(gf10( 102 k y ),2k6)(gf10( 102 k x 1nnn1n 1nnn1n nn 2 nn 4 1n nn1n nn1n           Jointly with (12), where .)10619()10619(g ,)10619()10619(f 1n1n n 1n1n n     Pattern 3: Assume 22 v9u9R  (13) Consider )i62)(i62(40  (14) Substituting (13), (14) in (3) and using the method of factorizations, we have 4)v3iu3)(i62()t2is3(  Equating real and imaginary parts, we have IJO JOURNALS Volume 07 | Issue 05 | May 2024 | https://ijojournals.com/index.php/m/index 5 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) DR. N.THIRUNIRAISELVI1* https://ijojournals.com/ Volume 07 || Issue 05|| May, 2024 || )]vu(uv4v3vu18u3[3t )]vu(uv24v2vu12u2[3s 4442244 4442243   In view of (2), we get )]vu(uv28vvu6u[3*6q )]vu(uv20v5vu30u6[3*6p )]vu(uv4v3vu18u3[3z )]vu(uv32v4vu24u4[3y )]vu(uv16v8vu48u8[3x 2242247 2242247 2242247 2242244 2242243      Pattern 4: It is worth to mention that the integer 40 may also be written as 222 )( )],(g2i),(2)][,(g2i),(f2[ 40    (15) where )()2(3),(g;2)(3),(f 2222   Substituting (13) & (15) in (3) and employing the method of factorization, we get )]v,u(iG)v,u(F)][,(g2i),(f2[ )( 3 t2is3 22 4    (16) where )vu(uv4)v,u(G vvu6u)v,u(F 22 4224   Equating the real and imaginary part of (16), one obtains )]v,u(F),(g)v,u(G),(f[ )( 3 t )]v,u(G),(g2)v,u(F),(f2[ )( 3 s 22 4 22 3       (17) Replacing u by u)( 22  , v by v)( 22  in (16), we have )]v,u(F),(g)v,u(G),(f[)(3t )]v,u(G),(g2)v,u(F),(f2[)(3s 3224 3223   (18) Also, from (13), )vu()(9R 22222  (19) From (2) and (18), it is seen that IJO JOURNALS Volume 07 | Issue 05 | May 2024 | https://ijojournals.com/index.php/m/index 6 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) DR. N.THIRUNIRAISELVI1* https://ijojournals.com/ Volume 07 || Issue 05|| May, 2024 || )]v,u(G),(g2)v,u(F),(f2)v,u(F),(g3)v,u(G),(f3[)(3*6q )]v,u(G),(g2)v,u(F),(f2)v,u(F),(g3)v,u(G),(f3[)(3*6p )]v,u(F),(g)v,u(G),(f)][v,u(G),(g2)v,u(F),(f2[)(3z )}]v,u(G)v,u(F){,(g2)}v,u(G)v,u(F){,(f2[)(3y )}]v,u(G)v,u(F){,(g2)}v,u(G)v,u(F){,(f2[)(3x 3223 3223 6227 3224 3224      which satisfy (1) along with (19). Pattern 5: Rewrite (3) as 1*R40t4s9 422  (20) Assume the integer 1 on the RHS of (1) as )qp( )pq2iqp)(pq2iqp( 1 22 2222    (21) Substituting (14), (21) in (20), we have )]q,p,v,u(iQ)q,p,v,u(P[ )qp( )i62(3 )t2is3( )]q,p(iK)q,p(J)][v,u(iG)v,u(F[ )qp( )i62(3 )t2is3( )v3is3( )qp( )pq2iqp( )i62()t2is3( 22 4 22 4 4 22 22             where )q,p(J)v,u(G)q,p(K)v,u(F)q,p,v,u(Q )q,p(K)v,u(G)q,p(J)v,u(F)q,p,v,u(P )vu(uv4)v,u(G;vvu6u)v,u(F pq2)q,p(K;qp)q,p(J 224224 22     Equating real and imaginary part of the above equation, we have )]q,p,v,u(Q)q,p,v,u(P3[ )qp( 3 t )]q,p,v,u(Q6)q,p,v,u(P2[ )qp( 3 s 22 4 22 3       IJO JOURNALS Volume 07 | Issue 05 | May 2024 | https://ijojournals.com/index.php/m/index 7 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) DR. N.THIRUNIRAISELVI1* https://ijojournals.com/ Volume 07 || Issue 05|| May, 2024 || Replacing u by (p2+ q2) A, v by (p2+ q2) B in the values of s, t and R, we have )BA()qp(9R )]q,p,B,A(Q)q,p,B,A(P3[)qp(81t )]q,p,B,A(Q6)q,p,B,A(P2[)qp(27s 22222 322 322    Conclusion: It has been shown that higher degree Diophantine equation with multiple variables may be solved through strategies like substitution and factorization by reducing it to a lesser degree solvable Diophantine equation. One may search for other choices of higher degree Diophantine equations with multiple variables for obtaining their respective integer solutions. References: [1] J.Sivasankari, Dr.R.Anbuselvi, “Integral solutions for the Diophantine Equation of Higher Degree with six Unknowns 822366 R)qp(800z3456yx  ”, Advances in Nonlinear Variational Inequalities, Vol 27(1), 2024. IJO JOURNALS Volume 07 | Issue 05 | May 2024 | https://ijojournals.com/index.php/m/index 8