Through a sufficiently developed introduction, this article begins by recalling the FPM (Fractional Power Model), \(A+Bt^m\) with positive t and m, and by specifying its contributions in terms of representativity, predictivity and application to \(CO_2\) concentration. Concerning the model representativity, its capacity to represent an indefinite number of internal dynamics, of different rapidity, is proven by the decomposition of \(t^m\) into its internal dynamics, and this, by using the synthesis transmittance of the integrator of non-integer order, m, whose step response is \(t^m\) for \(t>0\) . As for the model predictivity, its capacity to take into account all the past, by weighting it as appropriate, is proven by the predictive form with long memory of \({t}^{m}\) : this predictive form is established, here, by taking into account first an integer power, then by extending the results obtained to a non-integer one, which makes it possible to highlight the interest of a non-integer power in the consideration of the past. Beyond these fundamental aspects and although the FPM model was originally conceived for a viral spreading, the model is used here to represent and predict the evolution of \({CO}_2\) concentration in the atmosphere, where data are also cumulated. In order to lead a comparative study of predictivity, the article strives to compare the results obtained in prediction with three models having the same number of parameters: the FPM model, a polynomial model of degree 2 and an exponential model: the prediction errors (on a prediction horizon belonging to the past) clearly prove to be in favor of the FPM model which well illustrates that the greater the past taken into account, the better the predictivity. Being the most predictive, only the FPM model is then used as a predictor in the case of a prediction horizon belonging to the future; namely a horizon on 35 years, from 2022 to 2056, where, at the end, the relative dispersion of the predictions remains between \(-0.69\%\) and \(+0.59\%\) .
