I am dying to hear about it, since I always thought number theory was the Queen of Mathematics—the purest branch of mathematics—the one branch of mathematics which has NO applications!

——D. R. Hofstadter, *Gödel, Escher, Bach*

**Now that we are fitted** out with a sturdy tool box of arithmetic functions that we developed in the previous chapters, we turn our attention to the implementation of several fundamental algorithms from the realm of number theory. The number-theoretic functions discussed in the following chapters form a collection that on the one hand exemplifies the application of the arithmetic of large numbers and on the other forms a useful foundation for more complex number-theoretic calculations ...

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