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Gopalan, M.A.
- Observations on the Non-homogeneous Quintic Equation with Four Unknowns
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Authors
Affiliations
1 Department of Mathematics, Shrimati Indira Gandhi College, Trichy–620002, IN
1 Department of Mathematics, Shrimati Indira Gandhi College, Trichy–620002, IN
Source
International Journal of Mathematics Research, Vol 5, No 1 (2013), Pagination: 127-133Abstract
We obtain infinitely many non-zero integer quadruples (x, y, z,w) satisfying the quintic equation with four unknowns. xy+6z2 = (k2 + 5)n w5. Various interesting relations between the solutions and special numbers, namely, polygonal numbers, pyramidal numbers, Jacobsthal numbers, Jacobsthal-Lucas number, keynea numbers, Four Dimensional Figurative numbers and Five Dimensional Figurative numbers are exhibited.Keywords
Quintic Equation With Four Unknowns, Integral Solutions, 2- dimentional, 3-dimentional, 4-dimentional and 5-dimensional Figurative numbers.
References
- L.E. Dickson,History of Theory of Numbers,Vol.11,Chelsea Publishing company, New York (1952).
- L.J. Mordell, Diophantine equations, Academic Press, London(1969).
- Carmichael ,R.D.,The theory of numbers and Diophantine Analysis,Dover Publications, New York (1959)
- M.A. Gopalan & A.Vijayashankar, An Interesting Diophantine problem x3+y3=2z5 , Narosa Publishing House, Pp 1-6, 2010.
- M.A. Gopalan & A.Vijayashankar, Integral solutions of ternary quintic Diophantine equation x2+(2k+1)y2=z5 , International Journal of Mathematical Sciences 19(1-2), 165-169,(jan-june 2010)
- M.A. Gopalan,G.Sumathi & S.Vidhyalakshmi, Integral solutions of nonhomogeneous ternary quintic equation in terms of pells sequence x3+y3=xy(x+y)=2Z5, accepted for Publication in JAMS(Research India Publication)
- S. Vidhyalakshmi, K.Lakshmi and M.A.Gopalan, Observations on the homogeneous quintic equation with four unknowns x5-y5=2z5+5(x+y)(x2-y2)w2 ,accepted for Publication in International Journal of Multidisciplinary Research Academy(IJMRA)
- M.A. Gopalan & A.Vijayashankar, Integral solutions of non-homogeneous quintic equation with five unknowns xy-zw=R5 , Bessel J.Math.,1(1),23- 30,2011.
- M.A. Gopalan & A.Vijayashankar, solutions of quintic equation with five unknowns x4-y4=2(z2-w2)P3 ,accepted for Publication in International Review of Pure and Applied Mathematics.
- M.A. Gopalan, G. Sumathi & S.Vidhyalakshmi, On the non-homogenous quintic equation with five unknowns x3+y3=z3+w3+6T5, accepted for Publication in International Journal of Multidisciplinary Research Academy (IJMRA).
- Observations on the Non-homogenous Biquadratic Equation with Four Unknowns
Abstract Views :577 |
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Authors
Affiliations
1 Department of Mathematics, Shrimati Indira Gandhi College, Tiruchirappalli–2, IN
1 Department of Mathematics, Shrimati Indira Gandhi College, Tiruchirappalli–2, IN
Source
International Journal of Mathematics Research, Vol 5, No 1 (2013), Pagination: 135-140Abstract
We obtain infinitey many non-zero integer quadruples (x,y,z,w) satisfying the biquadratic equation with four unknowns 8(x3+y3)=(1+3k2)z3w.. Various interesting relations between the solutions and special numbers, namely, polygonal numbers, pyramidal numbers, Jacobsthal numbers, Jacobsthal- Lucas numbers are obtained.Keywords
Bi-quadratic Equation With Four Unknowns, Integral Solution, Special Numbers.References
- Dickson. I. E. ‘”History of the Theory of Numbers’’, Vol 2. Diophantine Analysis, New York, Dover, 2005.
- Mordell.L J. “ Diophantine Equations’’ Academic Press, New York,1969.
- Carmichael. R.D.” The Theory of numbers and Diophantine Analysis’’ ,New
- York, Dover, 1959.
- Gopalan.M.A., Manju Somanath and Vanitha.N.”Parametric Integral Solution of x2+y3=z4 ’’Acta ciencia Indica,Vol.XXXIIIM,No.4,1261-1265,2007.
- Gopalan.M.A., Pandichelvi.V.”On ternary quartic Diophantine equation x2+ky3=z 4 ’’Pacific-Asian Journal of mathematics,Vol.2, No.1-2,57- 62,2008.
- Gopalan.M.A. Manju somanath and Vanitha.N ,” Integral Solutions of x2+xy+y2 = (k2+3)nz<4’’ Pure and applied mathematika sciences, Vol. LXIX, No.1-2,149-152, 2009.
- Gopalan.M.A,Janaki.G, ”Observation on 2(x2-y2 )+4xy=z 4 ‘’ Acta ciencia indica,Vol.XXXVM,No2,445-448,2009.
- Gopalan.M.A., Sangeetha.G“Integral solutions of ternary quartic equation x2+y2=2xy+z4 ’’Antartica .J.Math.,7(1) 95-101,2010.
- Gopalan.M.A.,A.Vijayasankar,.,”Integral solutions of ternary biquadratic equation x2+3y2=z4 ’’Impact J.Sci.and Tech.Vol.4,No.3,47-51,2010.
- Gopalan,M.A,Vidhyalakshmi.S,Devibala.S,’’Ternary quartic Diophantine equation 24n+3 (x3-y3)=z4 ’’Impact J.Sci.and Tech,Vol4,No 3,57-60,2010.
- Janaki G. Gopalan.M.A.’’Observations on 3(x2-y2 )+9xy=z4 ’’Antartica .J. Math., 7(2),239-245,2010.
- Gopalan, M.A. and Sangeetha.G’,Integral solutions of ternary nonhomogeneous biquadratic equation x4+x2+y2-y=z2+z’’Acta ciencia Indica,VolXXXVIIM,No.4,799-803,2011.
- Gopalan,M.A,Vidhyalakshmi.Sumathi.G,’’On the ternary biquadratic nonhomogenous equation (2k+1)(x2+y2+xy)=z4 '' Indian journal of engineering,Vol.1,No1,37-40,2012.
- Gopalan,M.A,Sumathi.G,Vidhyalakshmi.S,’’Integral solutions of ternary biquadratic non-homogeneous equation (α+1)(x2+y2)+(2k+1)xy=z4 ’’JARCE, Vol. 6, No (2).97-98, 2012.
- Gopalan,M.A,Sumathi.G,Vidhyalakshmi.S,’’Integral solutions of ternary biquadratic non-homogeneous equation (k+1)(x2+y2)-(2k+1)xy=z 4 ’’Archimedes J.Math,Vol.3(1),67-71.2013.