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Solutions of cubic equations in quadratic fields

100%

Chakraborty K., Kulkarni M.

Acta Arithmetica

1999

tom 89

nr 1

37-43

Let K be any quadratic field with $𝓞_K$ its ring of integers. We study the solutions of cubic equations, which represent elliptic curves defined over ℚ, in quadratic fields and prove some interesting results regarding the solutions by using elementary tools. As an application we consider the Diophantine equation r+s+t = rst = 1 in $𝓞_K$. This Diophantine equation gives an elliptic curve defined over ℚ with finite Mordell-Weil group. Using our study of the solutions of cubic equations in quadratic fields we present a simple proof of the fact that except for the ring of integers of ℚ(i) and ℚ(√2), this Diophantine equation is not solvable in the ring of integers of any other quadratic fields, which is already proved in [4].

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Diophantine equations and class number of imaginary quadratic fields

80%

Cao Z., Dong X.

Discussiones Mathematicae - General Algebra and Applications

2000

tom 20

nr 2

199-206

Let A, D, K, k ∈ ℕ with D square free and 2 ∤ k,B = 1,2 or 4 and $μ_{i} ∈ {-1,1}(i = 1,2)$, and let $h(-2^{1-e}D)(e = 0 or 1)$ denote the class number of the imaginary quadratic field $ℚ(√(-2^{1-e}D))$. In this paper, we give the all-positive integer solutions of the Diophantine equation Ax² + μ₁B = K((Ay² + μ₂B)/K)ⁿ, 2 ∤ n, n > 1 and we prove that if D > 1, then $h(-2^{1-e}D) ≡ 0 (mod n)$, where D, and n satisfy $kⁿ - 2^{e+1} = Dx²$, x ∈ ℕ, 2 ∤ n, n > 1. The results are valuable for the realization of quadratic field cryptosystem.

Pell’s Equation

80%

Acewicz M., Pąk K.

Formalized Mathematics

2017

tom 25

nr 3

197-204

In this article we formalize several basic theorems that correspond to Pell’s equation. We focus on two aspects: that the Pell’s equation x2 − Dy2 = 1 has infinitely many solutions in positive integers for a given D not being a perfect square, and that based on the least fundamental solution of the equation when we can simply calculate algebraically each remaining solution. “Solutions to Pell’s Equation” are listed as item #39 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.

Ograniczanie wyników

1 Acta Arithmetica

1 Discussiones Mathematicae - General Algebra and Applications

1 Formalized Mathematics

1 Acewicz M.

1 Cao Z.

1 Chakraborty K.

1 Dong X.

1 Kulkarni M.

1 Pąk K.

1 2017

1 2000

1 1999

Wyniki wyszukiwania

Solutions of cubic equations in quadratic fields

Diophantine equations and class number of imaginary quadratic fields

Pell’s Equation