On the Diophantine equation ax2 + by2 = ckn
نویسندگان
چکیده
منابع مشابه
ON THE DIOPHANTINE EQUATION Ax2+22m =yn
Let h denote the class number of the quadratic field Q( √−A) for a square free odd integer A> 1, and suppose that n> 2 is an odd integer with (n,h)= 1 and m> 1. In this paper, it is proved that the equation of the title has no solution in positive integers x and y if n has any prime factor congruent to 1 modulo 4. If n has no such factor it is proved that there exists at most one solution with ...
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have a solution in positive integers (x, y, z) other than that given by x = 5, y = 7 and z = 4. A negative answer to this question follows from a classical result of Ljunggren [8], as recently refined by Cohn [4]: Theorem 1.1. Let the fundamental solution of the equation v2−Du2 = 1 be a + b √ D (i.e. (v, u) = (a, b) is the smallest positive solution). Then the only possible solutions of the equ...
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In this paper, we study the Diophantine equation x2 + C = 2yn in positive integers x, y with gcd(x, y) = 1, where n ≥ 3 and C is a positive integer. If C ≡ 1 (mod 4) we give a very sharp bound for prime values of the exponent n; our main tool here is the result on existence of primitive divisors in Lehmer sequence due Bilu, Hanrot and Voutier. When C 6≡ 1 (mod 4) we explain how the equation can...
متن کاملOn the Diophantine Equation
= c for some integers a, b, c with ab 6= 0, has only finitely many integer solutions. Stoll & Tichy proved more generally that if a, b, c ∈ Q and ab 6= 0, then for m > n ≥ 3, the above equation has only finitely many integral solutions x, y. Independently, Rakaczki established a more precise finiteness result on this binomial equation and extended this result to more general equations (see Acta...
متن کاملOn the Diophantine Equation
If a, b and n are positive integers with b ≥ a and n ≥ 3, then the equation of the title possesses at most one solution in positive integers x and y, with the possible exceptions of (a, b, n) satisfying b = a + 1, 2 ≤ a ≤ min{0.3n, 83} and 17 ≤ n ≤ 347. The proof of this result relies on a variety of diophantine approximation techniques including those of rational approximation to hypergeometri...
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ژورنال
عنوان ژورنال: Indagationes Mathematicae
سال: 2005
ISSN: 0019-3577
DOI: 10.1016/s0019-3577(05)80030-8