(Note that in second-order logic—and effectively set theory— + and × can be defined just in terms of Δ ; see page 1160 . … A form of arithmetic in which one allows induction but removes multiplication was considered by Mojzesz Presburger in 1929.

Around 1621 Wilhelm Schickard probably built a machine based on gears for doing simplified multiplications involved in Johannes Kepler 's calculations of the orbit of the Moon. But much more widely known were the machines built in the 1640s by Blaise Pascal for doing addition on numbers with five or so digits and in the 1670s by Gottfried Leibniz for doing multiplication, division and square roots. … The types of machines discussed so far all have the feature that they have to be physically rearranged or rewired in order to perform different calculations.

The repetition period for a generator with rule n  Mod[a n, m] is given (for a and m relatively prime) by MultiplicativeOrder[a, m] .