Explicit solutions of imaginary quadratic norm equations

All solutions to Thomas' family of Thue equations over imaginary quadratic number fields

We completely solve the family of relative Thue equations x − (t − 1)xy − (t + 2)xy − y = μ, where the parameter t, the root of unity μ and the solutions x and y are integers in the same imaginary quadratic number field. This is achieved using the hypergeometric method for |t| ≥ 53 and Baker’s method combined with a computer search using continued fractions for the remaining values of t.

Diophantine Equations and Class Numbers of Imaginary Quadratic Fields

Let A,D, K, k ∈ N with D square free and 2 | /k, B = 1, 2 or 4 and μi ∈ {−1, 1}(i = 1, 2), and let h(−21−eD)(e = 0 or 1) denote the class number of the imaginary quadratic field Q( √−21−eD). In this paper, we give the all-positive integer solutions of the Diophantine equation Ax +μ1B = K ( (Ay +μ2B)/K )n , 2 | / n, n > 1 and we prove that if D > 1, then h(−21−eD) ≡ 0(mod n), where D, and n sati...

Thomas' Family of Thue Equations Over Imaginary Quadratic Fields

Imaginary Quadratic

is called the 2-class field tower of k. If n is the minimal integer such that kn = kn+1, then n is called the length of the tower. If no such n exists, then the tower is said to be of infinite length. At present there is no known decision procedure to determine whether or not the (2-)class field tower of a given field k is infinite. However, it is known by group theoretic results (see [2]) that...

Explicit exact solutions for variable coefficient Broer-Kaup equations

Based on symbolic manipulation program Maple and using Riccati equation mapping method several explicit exact solutions including kink, soliton-like, periodic and rational solutions are obtained for (2+1)-dimensional variable coefficient Broer-Kaup system in quite a straightforward manner. The known solutions of Riccati equation are used to construct new solutions for variable coefficient Broer...