$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

8: Other Topics in Number Theory

• • Contributed by Wissam Raji
• Associate Professor and the Chairman (Mathematics) at American University of Beirut

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

This chapter discusses various topics that are of profound interest in number theory. Section 1 on cryptography is on an application of number theory in the field of message decoding, while the other sections on elliptic curves and the Riemann zeta function are deeply connected with number theory. The section on Fermat’s last theorem is related, through Wile’s proof of Fermat’s conjecture on the non-existence of integer solutions to $$x^n+y^n=z^n$$ for $$n>2$$, to the field of elliptic curves (and thus to section 2).

Professor Ramez Maalouf from Notre Dame University, Lebanon for his contribution to chapter 8.

Contributors

• Dr. Wissam Raji, Ph.D., of the American University in Beirut. His work was selected by the Saylor Foundation’s Open Textbook Challenge for public release under a Creative Commons Attribution (CC BY) license.