  The $F_{10.7cm}$ radio flux and the Sunspot Number are the most widely used
long-term indices of solar activity. They are strongly correlated, which led to
the publication of many proxy relations allowing to convert one index onto the
other. However, those existing proxies show significant disagreements, in
particular at low solar activity. Our aim is to bring a global clarification of
those many issues. We compute new polynomial regressions up to degree 4, in
order to obtain a more accurate proxy. We also study the role of temporal
averaging on the regression, and we investigate the issue of the all-quiet
$F_{10.7}$ background flux. Finally, we check for any change in the $F_{10.7}$
-- sunspot number relation over the entire period 1947 -- 2015. We find that,
with a $4^{th}$-degree polynomial, we obtain a more accurate proxy relation
than all previous published ones, and we derive a formula giving standard
errors. The relation is different for daily, monthly and yearly mean values,
and it proves to be fully linear for raw non-averaged daily data. By a simple
two-component model for daily values, we show how temporal averaging leads to
non-linear proxy relations. We also show that the quiet-Sun $F_{10.7}$
background is not absolute and actually depends on the duration of the spotless
periods. Finally, we find that the $F_{10.7cm}$ time series is inhomogeneous,
with an abrupt 10.5% upward jump occurring between 1980 and 1981. Our new proxy
relations show the importance of temporal scale for choosing the appropriate
proxy and the $F_{10.7}$ quiet-Sun background level. From historical evidence,
we conclude that the 1981 jump is most likely due to a unique change in the
$F_{10.7}$ scientific team and the data processing, and that the newly
re-calibrated sunspot number (version2) will probably provide the only possible
reference to correct this inhomogeneity.
