Diophantine equation X
 4+Y
 4 = 2(U
 4 + V
 4)

Izadi, Farzali; Nabardi, Kamran

doi:10.1515/ms-2015-0157

Cited by 6 publications

(10 citation statements)

References 11 publications

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“…This is the same DE (1) given in the Introduction for k = 6A 2 , that has been studied by some authors (see [1][2][3][4][5][6]). Then we may solve the DE (1) for all of the values k = 6A 2 , which rank of the elliptic curve (11) is positive.…”

Section: Now We Work Out Some Examplesmentioning

confidence: 54%

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

Abdolmalki¹,

Izadi²

2018

NNTDM

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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.

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Section: Now We Work Out Some Examplesmentioning

confidence: 54%

“…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%

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

Abdolmalki¹,

Izadi²

2018

NNTDM

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“…We use the fact that T (Q) is infinite to deduce that infinitely many specializations of E have the rank at least 5 over Q. This is done by using some elliptic curves of positive rank lying on T that we found in [8,9].…”

Section: Previous Workmentioning

confidence: 99%

“…In [8] Izadi and Nabardi found infinitely many integer solutions of this equation. Their method is based on the points of the elliptic curve y 2 = x 3 − 36x with a generator (−3, 9) explained as follows: Let P n = (x n , y n ), where P n = n · (−3, 9) (n ∈ N) is a point on the elliptic curve y 2 = x 3 − 36x, one gets A n = φ n 4 + 1296ψ n 8 + 864φ n ψ n 6 + 72φ n 2 ψ n 4 + 144ω n ψ n 5 −24φ n 3 ψ n 2 + 4φ n 2 ω n ψ n , D n = −864φ n ψ n 6 − φ n 4 − 1296ψ n 8 − 72φ n 2 ψ n 4 + 144ω n ψ n 5 +24φ n 3 ψ n 2 + 4φ n 2 ω n ψ n , B n = 4(φ n 2 + 36ψ n 4 )ω n ψ n ,…”

Section: Finding the Solutions Of The Equationmentioning

confidence: 99%

A family of elliptic curves with rank $$\ge 5$$ ≥ 5

Izadi

Nabardi

2015

Period Math Hung

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In this paper, we construct a family of elliptic curves with rank ≥ 5. To do this, we use the Heron formula for a triple (A 2 , B 2 , C 2 ) which are not necessarily the three sides of a triangle. It turns out that as parameters of a family of elliptic curves, these three positive integers A, B, and C, along with the extra parameter D satisfy the quartic Diophantine equationKeywords Diophantine equation · elliptic curve · Heron formula 1 IntroductionAs is well-known, the affine part of an elliptic curve E over a field K can be explicitly expressed by the generalized Weierstrass equation of the form E : y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 ,where a 1 , a 2 , a 3 , a 4 , a 6 ∈ K. In this paper we are interested in the case of K = Q. By the Mordell-Weil theorem [23], every elliptic curve over Q has a commutative group E(Q) which is finitely generated, i.e., E(Q) ∼ = Z r × E(Q) tors , where r is a nonnegative integer called the rank of E(Q) and E(Q) tors is the subgroup of elements of finite order called the torsion subgroup of E(Q).

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“…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%

Untitled

2018

NNTDM

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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.

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Diophantine equation X ⁴+Y ⁴ = 2(U ⁴ + V ⁴)

Abstract: Abstract. In this paper, the theory of elliptic curves is used for finding the solutions of the quartic Diophantine equation X 4 + Y 4 = 2(U 4 + V 4 ).

Cited by 6 publications

References 11 publications

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

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

A family of elliptic curves with rank $$\ge 5$$ ≥ 5

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Diophantine equation X 4+Y 4 = 2(U 4 + V 4)

Abstract: Abstract. In this paper, the theory of elliptic curves is used for finding the solutions of the quartic Diophantine equation X 4 + Y 4 = 2(U 4 + V 4 ).

Cited by 6 publications

References 11 publications

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

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

A family of elliptic curves with rank $$\ge 5$$ ≥ 5

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Diophantine equation X ⁴+Y ⁴ = 2(U ⁴ + V ⁴)