منابع مشابه
Exponential Diophantine Equations
1. Historical introduction. Many questions in number theory concern perfect powers, numbers of the form a b where a and b are rational integers with <7>1, 6>1. To mention a few: (a) Is it possible that for /zs>3 the sum of two 77th powers is an /7th power? (b) Is 8, 9 the only pair of perfect powers which differ by 1 ? (c) Is it possible that the product of consecutive integers, (x+l)(x + 2) .....
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We deal with a conjecture of Terai (1994) asserting that if a, b, c are fixed coprime integers with min(a, b, c) > 1 such that a+b = c for a certain odd integer r > 1, then the equation a + b = c has only one solution in positive integers with min(x, y, z) > 1. Co-operation man-machine is needed for the proof.
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We look at the relationships between class numbers of quadratic structures (orders and fields) and the solutions of exponential Diophantine equations. We conclude with necessary and sufficient conditions for a class group to have an element of a given order. 1. Notation and Preliminaries If D is a squarefree integer, then its discriminant is given by ( ) ( ) ≡ ≡/ = ∆ . 4 mod 1 if , 4 mod ...
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The first course is devoted to the basic setup of Diophantine approximation: we start with rational approximation to a single real number. Firstly, positive results tell us that a real number x has “good” rational approximation p/q, where “good” is when one compares |x − p/q| and q. We discuss Dirichlet’s result in 1842 (see [6] Course N◦2 §2.1) and the Markoff–Lagrange spectrum ([6] Course N◦1...
متن کاملOn the Exponential Diophantine Equation
Let a, b, c be fixed positive integers satisfying a2 + ab + b2 = c with gcd(a, b) = 1. We show that the Diophantine equation a2x+axby+b2y = cz has only the positive integer solution (x, y, z) = (1, 1, 1) under some conditions. The proof is based on elementary methods and Cohn’s ones concerning the Diophantine equation x2 + C = yn. Mathematics Subject Classification: 11D61
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ژورنال
عنوان ژورنال: Glasgow Mathematical Journal
سال: 1997
ISSN: 0017-0895,1469-509X
DOI: 10.1017/s0017089500032122