On the diophantine equation $x^{2}+5^{a}\cdot 11^{b}=y^{n}$
نویسندگان
چکیده
منابع مشابه
On the Diophantine Equation
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...
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= 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...
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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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In this remark, we use some properties of simple continued fractions of quadratic irrational numbers to prove that the equation x3 − 1 x− 1 = y − 1 y − 1 , x, y, n ∈ N, x > 1, y > 1, n > 3, 2 ∤ n has only the solutions (x, y, n) = (5, 2, 5) and (90, 2, 13). For any positive integer N with N > 2, let s(N) denote the number of solutions (x,m) of the equation (1) N = x − 1 x− 1 , x,m ∈ N, x ≥ 2, m...
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In this paper we study the equation x+7 = y, in integers x, y, m with m ≥ 3, using a Frey curve and Ribet’s level lowering theorem. We adapt some ideas of Kraus to show that there are no solutions to the equation with m composite and m > 15, and none with m prime and 11 ≤ m < 10.
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ژورنال
عنوان ژورنال: Functiones et Approximatio Commentarii Mathematici
سال: 2010
ISSN: 0208-6573
DOI: 10.7169/facm/1291903397