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As long as a branch of science offers an abundance of problems, so long it is alive; a lack of problems foreshadows extinction or the cessation of independent development.–David Hilbert, 1900

The subject of my talk is perhaps most directly indicated by simply asking two questions: first, is it harder to multiply than to add? and second, why?… I (would like to) show that there is no algorithm for multiplication computationally as simple as that for addition, and this proves something of a stumbling block.