On the exceptional set for binary Egyptian fractions

Details On the exceptional set for binary Egyptian fractions On the exceptional set for binary Egyptian fractions

2011-11-10

arxiv.org/abs/1111.2574v1

Mathematics

Number Theory

13 pages

Scientific paper

For fixed integer $a\ge3$, we study the binary Diophantine equation $\frac{a}n=\frac1x+\frac1y$ and in particular the number $E_a(N)$ of $n\le N$ for which the equation has no positive integer solutions in $x, y$. The asymptotic formula $$E_a(N)\sim C(a) \frac{N(\log\log N)^{2^{m-1}-1}}{(\log N)^{1-1/2^m}}$$ as $N$ goes to infinity, is established in this article, and this improves the best result in the literature dramatically. The proof depends on a very delicate analysis of the underlying group structure.

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