Abstract
Different authors have done analysis regarding sums of powers (Ref. no. 1,2 & 3), but systematic approach for solving Diophantine equations having sums of many bi-quadratics equal to a quartic has not been done before. In this paper we give methods for finding numerical solutions to equation (A) given above in section one. Next in section two, we give methods for finding numerical solutions for equation (B) given above. It is known that finding parametric solutions to biquadratic equations is not easy by conventional method. So the authors have found numerical solutions to equation (A) & (B) using elliptic curve theory.
Highlights
Elliptic Curves found any publications which deal systematically, with the subject of (k + n) bi-quadratics, where (n=3, 5)
Numerical solutions where k>9, but the authors are going to deal with solutions for k=1,2,3,4,5,6,7,8,9 for (k+5) bi-quadratics and for k=2,3,7,8,9 for (k+3) bi-quadratics
K=1 is omitted because a solution has been given by Jacobi & Madden in their paper on the equation,4 =4
Summary
Elliptic Curves found any publications which deal systematically, with the subject of (k + n) bi-quadratics, where (n=3, 5). Mention of k=4 & 6 is not made in section two because k=4 & 6 has elliptic curve, with rank zero, has no rational integer solutions. Refer to the elliptic curve tables mentioned in the reference section: TThee pppppppppp (XX, YY) oooo tthee eeeeeeeeeeeeeeeeeeee cccccccccc aaaaaaaaaa = (580 , 23368) , llllllllll tttt bbbbbbbbbb.
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