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Showing 1 to 4 of 4 for “"Egyptian fractions"”.
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On the Number of Representations of One as the Sum of Unit Fractions
The Egyptian Fractions of One problem (EFO), asks the following question: Given a positive integer n, how many ways can 1 be expressed as the sum of n non-increasing unit fractions? In this paper, we verify a result concerning the EFO problem for n=8, and show the computational complexity of the …
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Historic development of prime numbers
… Bone', `Rhine Papyrus' with an investigation of Egyptian fractions, and Euclid's Elements where the proof of the existence of an infinite number of primes was first given. The `Dark Ages' section has topics on non-European mathematicians who worked with prime numbers. The final section covers …
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Combinatorial aspects of synchronisation, percolation, and arithmetic
… Hetzel and Pappalardi on the number of Egyptian fractions. We show that for all $k \geq 2$, the number of integers $1 \leq a \leq n$ such that the equation $a/n = 1/m_1 + \dotsb + 1/m_k$ has a solution in integers $1 \leq m_1 \leq \dotsb \leq m_k$ is bounded above by $n^{1 - 1/2^{k-2} + …
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Topics in Probabilistic Combinatorics
… Finally, in Chapter 9, we are interested in Egyptian fractions. For a prime number $p$, we let $A_3(p)= | \{ m \in \mathbb{N}: \exists m_1,m_2,m_3 \in \mathbb{N}, \frac{m}{p}=\frac{1}{m_1}+\frac{1}{m_2}+\frac{1}{m_3} \} |$ be the number of fractions with denominator $p$ that can be written as …