On the Homogeneous Biquadratic Equation with 5 Unknowns (x2–y2)((4k-1)(x2+y2)-(4k-2)xy) = 2(4k-1)(p2–q2)z2
M. A. Gopalan1, K. Geetha2, Manju Somanath3
1M. A. Gopalan, Professor, Department of Mathematics, Shrimathi Indira Gandhi College, Trichy, (Tamil Nadu), India.
2K. Geetha, Assistant Professor, Department of Mathematics, Cauvery College for Women, Trichy, (Tamil Nadu), India.
3Manju Somanath, Assistant Professor, Department of Mathematics, National College, Trichy, (Tamil Nadu), India.
Manuscript received on June 27, 2015. | Revised Manuscript received on July 07, 2015. | Manuscript published on July 15, 2015. | PP: 6-10 | Volume-3 Issue-8, July 2015. | Retrieval Number: H0914073815/2015©BEIESP
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Abstract: The homogeneous biquadratic equation with five unknowns given by (x2–y2)((4k-1)(x2+y2)-(4k-2)xy) = 2(4k-1)(p2–q2)z2 is considered and analyzed for finding its non zero distinct integral solutions. Introducing the linear transformations x=u+v, y=u-v, p=2u+v, q=2u-v and employing the method of factorization different patterns of non zero distinct integer solutions of the equation under the above equation are obtained. A few interesting relations between the integral solutions and the special numbers namely Polygonal numbers, Pyramidal numbers, Centered Polygonal numbers, Centered Pyramidal numbers, Thabit-ibn-Kurrah number, Carol number, Mersenne number are exhibited.
Keywords: Homogeneous equation, Integral solutions, Polygonal numbers, Pyramidal numbers and Special numbers. 2010 Mathematics Subject Classification Code: 11D25