Mordell Diophantine Equations Pdf 13

Fermat's equation, xn + yn = zn with positive integer solutions, is an example of a Diophantine equation,[22] named for the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers x and y such that their sum, and the sum of their squares, equal two given numbers A and B, respectively:

Mordell Diophantine Equations Pdf 13

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Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation x2 + y2 = z2 are given by the Pythagorean triples, originally solved by the Babylonians (c. 1800 BC).[26] Solutions to linear Diophantine equations, such as 26x + 65y = 13, may be found using the Euclidean algorithm (c. 5th century BC).[27]Many Diophantine equations have a form similar to the equation of Fermat's Last Theorem from the point of view of algebra, in that they have no cross terms mixing two letters, without sharing its particular properties. For example, it is known that there are infinitely many positive integers x, y, and z such that xn + yn = zm where n and m are relatively prime natural numbers.[note 2]

Fermat's Last Theorem considers solutions to the Fermat equation: an + bn = cn with positive integers a, b, and c and an integer n greater than 2. There are several generalizations of the Fermat equation to more general equations that allow the exponent n to be a negative integer or rational, or to consider three different exponents.

Graduate School of Science and Technology, Hirosaki University,Hirosaki, 036-8561 Japan e-mail: komatsu@cc.hirosaki-u.ac.jpDepartment of Mathematics, Keio University, Yokohama, 223-8522 Japan e-mail: shiokawa@math.keio.ac.jpAbstract. We compute upper and lower bounds for the approximation ofcertain values ξ of hyperbolic and trigonometric functions by rationals x/ysuch that x, y satisfy Diophantine equations.We show that there are infinitely many coprime integers x, y such thaty ξ - x (log log y)/(log y)

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