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Abstract |
In order to understand what could be expected for Waring's problem (I.e. expressing integers as sums of a bounded number of s-th powers), Erd\H{o}s and R\'{e}nyi introduced a random model for sums of s-th powers of integers. We shall present the history of the model, going through the arithmetic modification proposed with Hennecart and Landreau, as well as the latest developments (work under progress with Cilleruelo, Lambert and Plagne) concerning the possibility to represent-almost surely - any large integer as a sums of $s+1$ pseudo $s$-th powers, one of them being drastically limited in size. A general underlying theme of the lecture is: how good or bad are probabilistic models to mimic real life? |
