Constructions of diagonal quartic and sextic surfaces with infinitely many rational points

In this paper, by using elliptic curves theory, we study the quartic Diophantine equation (DE) n i=1 a i x 4 i = n j=1 a j y 4 j , where a i and n ≥ 3 are fixed arbitrary integers. We try to transform this quartic to a cubic elliptic curve of positive rank. We solve the equation for some values of a i and n = 3, 4, and find infinitely many nontrivial solutions for each case in natural numbers, and show among other things, how some numbers can be written as sums of three, four, or more biquadrates in two different ways. While our method can be used for solving the equation for n ≥ 3, this paper will be restricted to the examples where n = 3, 4. Finally, we explain how to solve more general cases (n ≥ 4) without giving concrete examples to case n ≥ 5.

Section: Introductionmentioning

confidence: 99%

Untitled

2021

“…has already been studied by some authors. Four cases k = 1, k = 6r + 1, k = −8, and k = 2n 2 , where n is a congruent number, has been studied in [1][2][3][4][5][6].…”

Section: Introductionmentioning

confidence: 99%

“…where n ≥ 1, and T i , are rational numbers. This also solves the DE (1) for some values of k, which is discussed in Example 1. We conclude this introduction with a standard fact which is needed in this paper (see [8], page 37).…”

Section: Introductionmentioning

confidence: 99%

Untitled

2018

In this paper, elliptic curves theory is used for solving the quartic Diophantine equation X 4 + Y 4 = 2U 4 + n i=1 T i U 4 i , where n ≥ 1, and T i , are rational numbers. We try to transform this quartic to a cubic elliptic curve of positive rank, then get infinitely many integer solutions for the aforementioned Diophantine equation. We solve the above Diophantine equation for some values of n, T i , and obtain infinitely many nontrivial integer solutions for each case. We show among the other things that some numbers can be written as sums of some biquadrates in two different ways with different coefficients.

“…has already been studied by some authors. Four cases k = 1, k = 6r + 1, k = −8, and k = 2n 2 , where n is a congruent number, has been studied in [1][2][3][4][5][6].…”

Section: Introductionmentioning

confidence: 99%

“…where n ≥ 1, and T i , are rational numbers. This also solves the DE (1) for some values of k, which is discussed in Example 1.…”

Section: Introductionmentioning

confidence: 99%

On a class of quartic Diophantine equations of at least five variables

Abdolmalki¹,

Izadi²

2018

In this paper, elliptic curves theory is used for solving the quartic Diophantine equationi , where n ≥ 1, and T i , are rational numbers. We try to transform this quartic to a cubic elliptic curve of positive rank, then get infinitely many integer solutions for the aforementioned Diophantine equation. We solve the above Diophantine equation for some values of n, T i , and obtain infinitely many nontrivial integer solutions for each case. We show among the other things that some numbers can be written as sums of some biquadrates in two different ways with different coefficients.