The Pell Equation
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چکیده
Leonhard Euler called (1) Pell’s Equation after the English mathematician John Pell (1611-1685). This terminology has persisted to the present day, despite the fact that it is well known to be mistaken: Pell’s only contribution to the subject was the publication of some partial results of Wallis and Brouncker. In fact the correct names are the usual ones: the problem of solving the equation was first considered by Fermat, and a complete solution was given by Lagrange.
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The 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 کاملThe Pell Equation x 2 − ( k 2 − k ) y 2 = 2 t
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...
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In [Lem2003b] we have sketched the historical development of problems related to Legendre’s equations ar−bs = 1 and the associated Pell equation x−dy = 1 with d = ab. In [Lem2003c] we discussed certain “non-standard” ideas to solve the Pell equation. Now we move from the historical to the modern part: below we will describe the theory of the first 2-descent on Pell conics and explain its connec...
متن کاملPell ’ s equation
1 On the so–called Pell–Fermat equation 2 1.1 Examples of simple continued fractions . . . . . . . . . . . . . 2 1.2 Existence of integer solutions . . . . . . . . . . . . . . . . . . 5 1.3 All integer solutions . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 On the group of units of Z[ √ D] . . . . . . . . . . . . . . . . . 8 1.5 Connection with rational approximation . . . . . . . . . . ....
متن کاملPell’s Equation, Ii
In Part I we met Pell’s equation x2−dy2 = 1 for nonsquare positive integers d. We stated Lagrange’s theorem that every Pell equation has a nontrivial solution (an integral solution besides (±1, 0)) and saw what all solutions to Pell’s equation look like if there’s a nontrivial solution. As in Part I, “solution” means integral solution. Here we will prove Lagrange’s theorem in Section 2 and show...
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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...
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تاریخ انتشار 2011