20413 Elementary Number Theory

Credits: 4 intermediate credits in Mathematics

Prerequisites: none

Recommended: Having accumulated about 15 credits in Mathematics

The course, based on a translation (by Naomi Shaked-Monderer) of Elementary Number Theory (5th ed.), by D.M. Burton (McGraw Hill, 2002), was developed by Abraham Ginzburg, Israel Friedman and Yonutz Stanchescu.

Number theory is an age-old topic that challenged the imagination of the foremost mathematicians throughout the generations and has attracted amateurs as well. Many number theory problems, which can be easily understood by laymen, require sophistication and intense efforts to solve them, and a multitude of open questions still remain.

The course presents basic topics in number theory in a straightforward manner that does not require advanced Mathematics knowledge. The historical notes woven into the text add flavor and the abundance of problems increases the challenge. The course, which may be of interest to Mathematics teachers as well as to students who wish to major in Mathematics or in Science, does not require specific mathematical knowledge. Nonetheless, some mathematical sophistication and calculation skills are necessary. Accordingly, students are urged to take at least three Mathematics courses prior to enrolling in this course.

Topics: Some preliminary considerations; Divisibility theory in the integers; Primes and their distribution; The theory of congruences; Fermatís theorem; Number-theoretic functions; Eulerís generalization of Fermatís theorem; Primitive roots and indices; The quadratic reciprocity law; Perfect numbers; The Fermat conjecture; Representation of integers as sums of squares; Fibonacci numbers; Continued fractions; Some 20th-century developments.