A Note on the Negative Pell Equation
نویسندگان
چکیده
We provide a new elementary proof of a criterion, given in earlier work, for the solvability of the Pell equation x2 −Dy2 = −1 where D is any positive nonsquare integer. Mathematics Subject Classification: Primary 11D09; 11A55; Secondary 11R11; 11R29
منابع مشابه
A RESEARCH NOTE ON THE SECOND ORDER DIFFERENTIAL EQUATION
Let U(t, ) be solution of the Dirichlet problem y''+( t-q(t))y= 0 - 1 t l y(-l)= 0 = y(x), with variabIe t on (-1, x), for fixed x, which satisfies the initial condition U(-1, )=0 , (-1, )=1. In this paper, the asymptotic representation of the corresponding eigenfunctions of the eigen values has been investigated . Furthermore, the leading term of the asymptotic formula for ...
متن کاملHigher Descent on Pell Conics. I. from Legendre to Selmer
The theory of Pell’s equation has a long history, as can be seen from the huge amount of references collected in Dickson [Dic1920], from the two books on its history by Konen [Kon1901] and Whitford [Whi1912], or from the books by Weber [Web1939], Walfisz [Wal1952], Faisant [Fai1991], and Barbeau [Bar2003]. For the better part of the last few centuries, the continued fractions method was the und...
متن کاملA Note on Stability of an Operator Linear Equation of the Second Order
and Applied Analysis 3 If p, 1/q ∈ Z, then solutions f : N0 → Z of the difference equation 1.7 are called the Lucas sequences see, e.g., 24 ; in some special cases they are given specific names; that is, the Fibonacci numbers p −1, q −1, f 0 0, and f 1 1 , the Lucas numbers p −1, q −1, f 0 2, and f 1 1 , the Pell numbers p −2, q −1, f 0 0, and f 1 1 , the Pell-Lucas or companion Lucas numbers p...
متن کاملSequences related to the Pell generalized equation
We consider sequences of the type An = 6An−1 − An−2, A0 = r, A1 = s (r and s integers) and show that all sequences that solve particular cases of the Pell generalized equation are expressible as a constant times one of four particular sequences of the same type. Let α = 3 + 2 √ 2, β = 3 − 2 √ 2 be the roots of the polynomial x − 6x+ 1. Note that α+β = 6, αβ = 1, α−β = 4 √ 2. Also let γ = 1+ √ 2...
متن کاملThe Pell Equation x 2 − ( k 2 − k ) y 2 = 2 t Ahmet
Let k, t, d be arbitrary integers with k ≥ 2, t ≥ 0 and d = k − k. In the first section we give some preliminaries from Pell equations x − dy = 1 and x − dy = N , where N be any fixed positive integer. In the second section, we consider the integer solutions of Pell equations x − dy = 1 and x − dy = 2. We give a method for the solutions of these equations. Further we derive recurrence relations...
متن کامل