MTH 210 Number theory Dr Anthony Bonato Ryerson

MTH 210 Number theory Dr. Anthony Bonato Ryerson University

Number theory • study of numbers, usually, integers • primes • later: congruences and Diophantine equations

Divisors • an integer m is a divisor of n if n = pm for some integer p • examples: – divisors of 10: 1, 2, 5, 10 – divisors of 16: 1, 2, 4, 8, 16 – divisors of 19: 1, 19

Primes • 19 is a special number – only divisors are 1, 19 • integer n > 1 is prime if its only divisors are 1 and n • NB: 1 is not prime

Primes up to 100 • 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97…

Primes never end Key fact: there are infinitely many primes.

Other kinds of numbers • rationals: p/q, p, q integers • reals: numbers with a decimal expansion • complex numbers: a + bi, where a, b reals, i 2 = -1

Irrational numbers • a real number is irrational if it is not rational – that is, it isn’t a fraction of integers

An irrational number •

Other irrationals • irrational, but much tougher to prove: π, e • no one knows if π + e is irrational!

Exercises