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Number theory is at once the most ancient of mathematical disciplines and one of the most modern. Its roots go back nearly as far as written records of civilization, and yet the most famous recent mathematical result (the proof of Fermat's Last Theorem), as well as some of the most important open problems of mathematics, are solidly within the tradition of number theory. Until recently, number theory was thought to be the purest part of mathematics, having no applications except to other fields of math. The situation has changed with the rise of public-key cryptography, which is now essential to electronic commerce worldwide.

This course begins by developing the elementary concepts of number theory such as prime numbers, divisibility, congruences, modular arithmetic, and Diophantine equations (equations to be solved in integers). For the most part we will study relationships among numbers for their own sake, but as a result we will be able to understand the two pillars of public-key cryptography: the RSA cryptosystem and Diffie-Hellman key exchange. The second half of the course will tackle quadratic reciprocity, approximation of irrational numbers by rational numbers, and a few very natural Diophantine equations.

The core material does not require any background beyond high-school algebra, but we assume a modest familiarity with mathematical reasoning such as might be provided by a course on sequences and series, like Haverford's Math 115. Facility with formal proofs will not be assumed, nor will it be taught; this is not a proof-writing course. Instead, we will emphasize the process of mathematical discovery: finding patterns in data and formulating precise conjectures.

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