منابع مشابه
The Pell Equation in Quadratic Fields
where 7 is a given integer of a quadratic field F, and integral solutions £, 77 are sought in F. It has been shown that equation (1) has an infinite number of solutions if and only if 7 is not totally negative when F is a real field, and 7 is not the square of an integer of F when F is imaginary. We now obtain the following result : Let 7 be such that equation (1) has an infinite number of solu...
متن کاملThe Pell Equation
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...
متن کاملPell ’ s Equation
An arbitrary quadratic diophantine equation with two unknowns can be reduced to a Pell-type equation. How can such equations be solved? Recall that the general solution of a linear diophantine equation is a linear function of some parameters. This does not happen with general quadratic diophantine equations. However, as we will see later, in the case of such equations with two unknowns there st...
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 کاملSolutions of the Cubic Fermat Equation in Quadratic Fields
We give necessary and sufficient conditions on a squarefree integer d for there to be non-trivial solutions to x + y = z in Q( √ d), conditional on the Birch and Swinnerton-Dyer conjecture. These conditions are similar to those obtained by J. Tunnell in his solution to the congruent number problem.
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ژورنال
عنوان ژورنال: Bulletin of the American Mathematical Society
سال: 1943
ISSN: 0002-9904
DOI: 10.1090/s0002-9904-1943-07934-7