Diophantine approximations with Fibonacci numbers
نویسندگان
چکیده
منابع مشابه
Diophantine quadruples and Fibonacci numbers
A Diophantine m-tuple is a set of m positive integers with the property that product of any two of its distinct elements is one less then a square. In this survey we describe some problems and results concerning Diophantine m-tuples and their connections with Fibonacci numbers.
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We show that there are only finitely many Diophantine quadruples, that is, sets of four positive integers {a1, a2, a3, a4} such that aiaj +1 is a square for all 1 ≤ i < j ≤ 4, consisting of Fibonacci numbers.
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Let FL = {1, 2, 3, 4, 5, 7, 8, 11, 13, 18, 21, . . .} be the set consisting of all Fibonacci and Lucas numbers with positive subscripts. We find all triples (a, b, c) of positive integers a < b < c such that ab + 1, ac+ 1, bc+ 1 are all members of FL.
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One can prove the following three propositions: (1) For all natural numbers m, n holds gcd(m,n) = gcd(m, n + m). (2) For all natural numbers k, m, n such that gcd(k, m) = 1 holds gcd(k,m · n) = gcd(k, n). (3) For every real number s such that s > 0 there exists a natural number n such that n > 0 and 0 < 1 n and 1 n ¬ s. In this article we present several logical schemes. The scheme Fib Ind conc...
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This article centres around the contributions of the author and therefore, it is confined to topics where the author has worked. Between these topics there are connections and we explain them by a result of Liouville in 1844 that for an algebraic number α of degree n ≥ 2, there exists c > 0 depending only on α such that | α− p q |> c qn for all rational numbers p q with q > 0. This inequality i...
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ژورنال
عنوان ژورنال: Journal de Théorie des Nombres de Bordeaux
سال: 2013
ISSN: 1246-7405,2118-8572
DOI: 10.5802/jtnb.846