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The $F_{10.7cm}$ radio flux and the Sunspot Number are the most widely usedlong-term indices of solar activity. They are strongly correlated, which led tothe publication of many proxy relations allowing to convert one index onto theother. However, those existing proxies show significant disagreements, inparticular at low solar activity. Our aim is to bring a global clarification ofthose many issues. We compute new polynomial regressions up to degree 4, inorder to obtain a more accurate proxy. We also study the role of temporalaveraging 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 relationthan all previous published ones, and we derive a formula giving standarderrors. 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 simpletwo-component model for daily values, we show how temporal averaging leads tonon-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 spotlessperiods. 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 proxyrelations show the importance of temporal scale for choosing the appropriateproxy 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 newlyre-calibrated sunspot number (version2) will probably provide the only possiblereference to correct this inhomogeneity.

Is the $F_{10.7cm}$ -- Sunspot Number relation linear and stable ?