PMATH340 Lecture 2: PMATH 340 lecture notes-2

• School: University of Waterloo
• Department: Pure Mathematics
• Course: PMATH340
• Professor: Kathryn Hare

Document Summary

This is a course on number theory, undoubtedly the oldest mathematical discipline known to the world. These may be integers, like 2,0 or 7, or rational numbers like 1/3 or 7/9, or algebraic. 2 or i, or transcendental numbers like e or . There are many interesting questions that one might ask about numbers. In search for answers to these questions mathematicians unravel their interesting properties, some of which are quite profound. Here are several curious facts about prime numbers: for x > 1, there are approximately x/logx prime numbers (prime number. Theorem, 1896): every odd number > 5 can be expressed as a sum of three primes (helfgott"s. Theorem, 2013): given a positive integer d, there exist distinct prime numbers p1, p2, . , pd which form an arithmetic progression (green-tao theorem, 2004). Despite the simplicity of their formulations, all of these results are highly non- trivial and their proofs reside on some deep theories.