On the Diophantine equation in the form that a sum of cubes equals a sum of quintics

Abstract

In this paper, the elliptic curves theory is used for solving the Diophantine equations a(X ′, where n, m ∈ N∪{0}, and, a, b = 0, a i , b i , are fixed arbitrary rational numbers. We solve the Diophantine equation for some values of n, m, a, b, a i , b i , and obtain nontrivial integer solutions for each case. By our method, we may find infinitely many nontrivial integer solutions for the Diophantine equation for every n, m, a, b, a i , b i , and show among the other things that how sums of some 5th powers can be written as sums of some cubics.

Remark on a Paper by Izadi and Baghalaghdam about Cubes and Fifth Powers Sums

“…has a rational point (s, w) = (1, 0). Hence we can parametrize rational points on C and integer solutions to (2). That is to say we have:…”

“…and obtain integer solutions, for example: 8 5 + 6 5 + 14 5 = (−110) 3 + 124 3 + 14 3 , 128122 5 +(−79524) 5 +48598 5 = 359227580 3 +(−251874598) 3 +107352982 3 . However, no positive solutions are presented in their paper [2]. In this paper, we refine their method to find positive solutions to (2).…”

“…However, no positive solutions are presented in their paper [2]. In this paper, we refine their method to find positive solutions to (2). Consider the Diophantine equation (2).…”

“…In this paper, we refine their method to find positive solutions to (2). Consider the Diophantine equation (2). Let:…”

“…with parameters x 1 , α, β ∈ Q. If we get a rational point (t, v) on C, we can compute a rational solution to (2) (see [2]).…”