Amicable Numbers and the Bilinear Diophantine Equation
نویسندگان
چکیده
منابع مشابه
On amicable numbers
Let A (x) denote the set of integers n≤ x that belong to an amicable pair. We show that #A (x)≤ x/e √ logx for all sufficiently large x.
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Define a quasi-amicable pair as a pair of distinct natural numbers each of which is the sum of the nontrivial divisors of the other, e.g., {48, 75}. Here nontrivial excludes both 1 and the number itself. Quasi-amicable pairs have been studied (primarily empirically) by Garcia, Beck and Najar, Lal and Forbes, and Hagis and Lord. We prove that the set of n belonging to a quasi-amicable pair has a...
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In this paper, we improve on several earlier attempts to show that the reciprocal sum of the amicable numbers is small, showing this sum is < 215.
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In a first article of this title, new procedures were described to compute many amicable numbers by “breeding” them in several generations. An extensive computer search was later performed (in 1988), and demonstrated the remarkable effectiveness of this breeding method: the number of known amicable pairs was easily quadrupled by this search. As we learnt recently (1999) from the internet, Peder...
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...
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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 1968
ISSN: 0025-5718
DOI: 10.2307/2004776