Denominators of Egyptian Fractions
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چکیده
منابع مشابه
. N T ] 8 A pr 1 99 8 DENSE EGYPTIAN FRACTIONS
Every positive rational number has representations as Egyptian fractions (sums of reciprocals of distinct positive integers) with arbitrarily many terms and with arbitrarily large denominators. However, such representations normally use a very sparse subset of the positive integers up to the largest denominator. We show that for every positive rational there exist representations as Egyptian fr...
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The Rhind Papyrus of Ahmes [47] (see also [34, 63]) is one of the oldest known mathematical manuscripts, dating from around 1650 B.C. It contains among other things, a list of expansions of fractions of the form 2 n into sums of distinct unit fractions, that is, fractions with numerator 1. Examples of such expansions are 2 35 = 1 30 + 1 42 and 2 63 = 1 56 + 1 72 . More generally, one can consid...
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Abstract. In this paper we examine the subset FQ,odd of Farey fractions of order Q consisting of those fractions whose denominators are odd. In particular, we consider the frequencies of the values taken on by ∆ = qa − aq where a/q < a/q are consecutive in FQ,odd. After proving an asymptotic result for these frequencies, we generalize the result to the subset of elements of FQ,odd formed by res...
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It is well known that the ancient Egyptians represented each fraction as a sum of unit fractions – i.e., fractions with unit numerators; this is how they, e.g., divided loaves of bread. What is not clear is why they used this representation. In this paper, we propose a new explanation: crudely speaking, that the main idea behind the Egyptian fractions provides an optimal way of dividing the loa...
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تاریخ انتشار 1972