A Note on the Number of Egyptian Fractions
A Note on the Number of Egyptian Fractions
Refining an estimate of Croot, Dobbs, Friedlander, Hetzel and Pappalardi, we show that for all $k \geq 2$, the number of integers $1 \leq a \leq n$ such that the equation $a/n = 1/m_1 + \dotsc + 1/m_k$ has a solution in positive integers $m_1, \dotsc, m_k$ is bounded above by $n^{1 - 1/2^{k-2} + o(1)}$ as $n$ goes to infinity. The proof is elementary.
2 pages
- University of Cambridge United Kingdom
- Department of Pure Mathematics and Mathematical Statistics University of Cambridge United Kingdom
- The University of Texas System United States
sums of arithmetic functions, Counting solutions of Diophantine equations, Mathematics - Number Theory, Rational numbers as sums of fractions, FOS: Mathematics, Egyptian fractions, Number Theory (math.NT), induction, sums of unit fractions
sums of arithmetic functions, Counting solutions of Diophantine equations, Mathematics - Number Theory, Rational numbers as sums of fractions, FOS: Mathematics, Egyptian fractions, Number Theory (math.NT), induction, sums of unit fractions
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