  We study solvability of the Diophantine equation \begin{equation*}
\frac{n}{2^{n}}=\sum_{i=1}^{k}\frac{a_{i}}{2^{a_{i}}}, \end{equation*} in
integers $n, k, a_{1},\ldots, a_{k}$ satisfying the conditions $k\geq 2$ and
$a_{i}<a_{i+1}$ for $i=1,\ldots,k-1$. The above Diophantine equation (of
polynomial-exponential type) was mentioned in the monograph of Erd\H{o}s and
Graham, where several questions were stated. Some of these questions were
already answered by Borwein and Loring. We extend their work and investigate
other aspects of Erd\H{o}s and Graham equation. First of all, we obtain the
upper bound for the value $a_{k}$ given in terms of $k$ only. This mean, that
with fixed $k$ our equation has only finitely many solutions in $n,
a_{1},\ldots, a_{k}$. Moreover, we construct an infinite set $\cal{K}$, such
that for each $k\in\cal{K}$, the considered equation has at least five
solutions. As an application of our findings we enumerate all solutions of the
equation for $k\leq 8$. Moreover, by applying greedy algorithm, we extend
Borwein and Loring calculations and check that for each $n\leq 10^4$ there is a
value of $k$ such that the considered equation has a solution in integers
$n+1=a_{1}<a_{2}<\ldots <a_{k}$. Based on our numerical calculations we
formulate some further questions and conjectures.
