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COUNTING THE NUMBER OF SOLUTIONS TO THE ERDŐS–STRAUS EQUATION ON UNIT FRACTIONS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 94 / Issue 1 / February 2013
- Published online by Cambridge University Press:
- 15 May 2013, pp. 50-105
- Print publication:
- February 2013
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- Article
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For any positive integer
$n$, let 
$f(n)$ denote the number of solutions to the Diophantine equation with
$$\begin{eqnarray*}\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\end{eqnarray*}$$
$x, y, z$ positive integers. The Erdős–Straus conjecture asserts that 
$f(n)\gt 0$ for every 
$n\geq 2$. In this paper we obtain a number of upper and lower bounds for 
$f(n)$ or 
$f(p)$ for typical values of natural numbers 
$n$ and primes 
$p$. For instance, we establish that These upper and lower bounds show that a typical prime has a small number of solutions to the Erdős–Straus Diophantine equation; small, when compared with other additive problems, like Waring’s problem.
$$\begin{eqnarray*}N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\ll \displaystyle \sum _{p\leq N}f(p)\ll N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\log \log N.\end{eqnarray*}$$
