Miniconference on the Mathematics of Computation 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, i2 = -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 Key fact: 2 is irrational.

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

Exercises