On The Non Homogeneous Heptic Equation With Five Unknowns (x2 - y2)(3x2 + 3y2 -5xy) = 5 (X2 –Y2) z5

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Year:
2016
Type of Publication:
Article
Keywords:
The Diophantine Equation, Heptic Equation, Integral Solutions, Special Numbers, a Few Interesting Relation
Authors:
Dr. P. Jayakumar; V. Pandian; J. Meena
Journal:
IJISM
Volume:
4
Number:
2
Pages:
98-101
Month:
March
ISSN:
2347-9051
Abstract:
Five different methods of the non-zero non-negative solutions of non- homogeneous Heptic Diophantine equation (x2 - y2)(3x2 + 3y2 -5xy) = 5 (X2 –Y2) z5 are observed. Introducing the linear transformation x = u + v, y = u – v, X =3u + u, Y= 3u – v , u  v 0 in (x2 - y2)(3x2 + 3y2 -5xy) = 5 (X2 –Y2) z5,it reduces to u2 +11v2 = 15z5. We are solved the above equation through various choices and the different methods of solutions which are satisfied it. Some interesting relations among the special numbers and the solutions are exposed. The following notations are used: Pnm: Pyramid number of rank n with size m- Tn,m : Polygonal number of rank n with size m- Ga : Gnomonic number of rank a- Pa : Pronic number of rank a- :Fourth and fifth dimensional figurate number of r, whose generating polygon is a Triangle- : Fourth and fifth dimensional figurate number of rank r, whose generating polygon is a Square - : Fourth and fifth dimensional figurate number of rank r, whose generating polygon is a Pentagon – : Fourth and fifth dimensional figurate number of rank r, whose generating polygon is a Hexagon- : Fourth and fifth dimensional figurate number of rank r, whose generating polygon is a Heptagon - : Fourth and fifth dimensional figurate number of rank r, whose generating polygon is a Octagon. 2010 Mathematics Subject Classification: 11D09

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