Number theory

All of these projects are related to number theory. Number theory may at first sound like topics that are only of interest to pure mathematicians, but engineers use many of the results from number theory in applications, including

• Hash functions for fast access and retrieval of data stored in large arrays.
• Pseudo-random numbers are used in simulations by generating sequences of numbers that, for all appearences, appear to be random, but are in fact reproducable.
• Binary, decimal and hexadecimal representations of integers and real numbers.
• Fast arithmetic operations such as multiplying two $n$-digit integers.
• Fast calculations of operations like ipow( b, e ) % n used in cryptography where ipow( b, e ) calculates $b^e$ for integers $b$ and $e$.
• Public-key cryptography.

The last topic is based heavily on the work of G.H. Hardy, who said at one point "No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world." He was wrong.

Projects

1. Sieve of Eratosthenes
2. Integer roots
3. Greatest common divisor and least common multiple
4. Iterative primality testing
5. The kth prime number