On the Non-Homogeneous Quintic Equation with Seven Unknowns
International Journal of Innovative Science and Modern Engineering (IJISME) ISSN: 2319-6386, Volume-3 Issue-6, May 2015 On the Non-Homogeneous Quintic Equation with Seven Unknowns S. Vidhyalakshmi, M. A. Gopalan, K. Lakshmi by . A few Abstract— We obtain infinitely many non-zero integer solutions (,,,,,,)x y z w X Y T satisfying the non- homogeneous relations between the solutions and the special numbers are presented quintic equation with seven unknowns given by Initially, the following two sets of solutions in 2 2 2 2 2 2 3 . Various xyx()()() y zwz w X YT (,,,,,,)x y z w X Y T satisfy the given equation: interesting relations between the solutions and special numbers, 2 2 2 namely, polygonal numbers, Pyramidal numbers, Stella (2k k p ,2 k k p , p 2 k k , , Octangular numbers, Octahedral numbers,, Jacobsthal number, 22 Jacobsthal-Lucas number, keynea number, Centered pyramidal p2 k k ,2 k ,4 k ,2 k ) numbers are presented (2k2 3 k p 1,2 k 2 3 k p 1,2 p k 2 k , Index Terms— Centered pyramidal numbers, Integral p2 k22 k ,2 k 1,4 k 4 k 1,2 k 1) solutions, Non-homogeneous Quintic equation, Polygonal numbers, Pyramidal numbers However we have other patterns of solutions, which are MSC 2010 Mathematics subject classification: 11D41. illustrated below: Notations: II. METHOD OF ANALYSIS Tmn, - Polygonal number of rank n with size m The Diophantine equation representing the non- SO - Stella Octangular number of rank n homogeneous quintic equation is given by n (1) Jn - Jacobsthal number of rank of Introduction of the transformations KY - Keynea number of rank n xupyupz ,,,, pvwpv CP - Centered hexagonal pyramidal number of rank n,6 X u v, Y u v (2) m in (1) leads to Pn - Pyramidal number of rank with size u2 v 2 T 3 (3) OH - Octahedral number of rank n The above equation (3) is solved through different approaches j - Jacobsthal-Lucas number of rank and thus, one obtains different sets of solutions to (1) n A.
[Show full text]