A Fibonacci-like Sequence of Composite Numbers
نویسندگان
چکیده
منابع مشابه
A Fibonacci-like Sequence of Composite Numbers
In 1964, Ronald Graham proved that there exist relatively prime natural numbers a and b such that the sequence {An} defined by An = An−1 +An−2 (n ≥ 2;A0 = a,A1 = b) contains no prime numbers, and constructed a 34-digit pair satisfying this condition. In 1990, Donald Knuth found a 17-digit pair satisfying the same conditions. That same year, noting an improvement to Knuth’s computation, Herbert ...
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An integer sequence (xn)n≥0 is said to be Fibonacci-like if it satisfies the binary recurrence relation xn = xn−1 + xn−2, n ≥ 2. We construct a new type of Fibonacci-like sequence of composite numbers. 1 The problem and previous results In this paper we consider Fibonacci-like sequences, that is, sequences (xn) ∞ n=0 satisfying the binary recurrence relation xn = xn−1 + xn−2, n ≥ 2. (1)
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We find a new Tribonacci-like sequence of positive integers 〈x0, x1, x2, . . .〉 given by xn = xn−1 + xn−2 + xn−3 , n ≥ 3, and gcd(x0, x1, x2) = 1 that contains no prime numbers. We show that the sequence with initial values x0 = 151646890045, x1 = 836564809606, x2 = 942785024683 is the current record in terms of the number of digits.
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The Farey sequence is an old and famous set of fractions associated with the integers. We here show that if we form a Farey sequence of Fibonacci Numbers, the properties of the Farey sequence are remarkably preserved (see [2]). In fact we find that with the new sequence we are able to observe and identify "points of symmetry/' "intervals," "generating fractions" and "stages." The paper is divid...
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ژورنال
عنوان ژورنال: The Electronic Journal of Combinatorics
سال: 1999
ISSN: 1077-8926
DOI: 10.37236/1476