Divisibility and Primes ICS 6 D Sandy Irani

Evenly Divides • x evenly divides y if y =m·x for some integer m – Denoted: x|y – y is an integer multiple (or just “multiple”) of x – x is a factor of y

Primes and Composites • A number p is prime if – p is an integer and p > 1 – The only (positive) factors of p are 1 and p. • If n > 1 is not prime, it is composite – n has a positive factor other than 1 or n.

Fundamental Theorem of Arithmetic • Every integer n > 1 can be expressed as a product of primes. Prime factorization is unique up to ordering. – Prime factorization of n usually written in nondecreasing order: 124 =

Fundamental Theorem of Arithmetic – Prime factorization of 360:

Exponential Notation for Prime Factorization • Write each distinct prime in the prime factorization in increasing order. • The multiplicity of a prime (# of times it appears) is written in the exponent. 124 = 360 =

Prime Factorization and Divisibility • If you have the prime factorization for x and y, can easily determine if x divides y: • Example: does 90 divide 1260? • 1260 = 22 · 32 · 5 · 7 • 90 = 2 · 32 · 5

Prime Factorization and Divisibility • 1260 = 22 · 32 · 5 · 7 • 40 = 23· 5 • 22 = 2 · 11

LCM and GCD • Two positive integers, x and y. – The least common multiple of x and y, lcm(x, y), is the smallest integer that is a multiple of x and a multiple of y. – The greatest common divisor of x and y, gcd(x, y) is the largest integer that divides both x and y. • x and y are relatively prime if gcd(x, y) = 1

Prime Factorization and LCM • If you have the prime factorization for x and y, can easily determine if lcm(x, y): 44 = 22 · 11 90 = 2 · 32 · 5

Prime Factorization and GDC • If you have the prime factorization for x and y, can easily determine if gcd(x, y): 1320 = 23 · 5 · 11 1800 = 22 · 32 · 5

Factoring: Brute-force algorithm • Input: integer N > 1 • Output: (x, y) such that x > 1, y > 1 and xy = N or “Prime” if N is prime • For x = 2 to N-1 – If x evenly divides N • Return(x, N/x) • Return(“Prime”)

Factoring: slightly better algorithm •

Factoring and Primality Testing • Factoring • Input: integer N > 1 • Output: – If N is prime: • “Prime” – If N is composite: • x > 1, y > 1, xy = N Hard: no known algorithm that Runs in time (# digits)d • Primality Testing • Input: integer N > 1 • Output: – If N is prime: • “Prime” – If N is composite: • “Composite” Easy: can be solved in time (# digits)2 (probabilistic)

Finding Prime Numbers • “Find a 200 -digit prime number” • How to do this? • Repeat: – Pick a random 200 digit number N – Apply primality test to N • If N is prime, return(N) • If N is composite, continue. The number of trials until success depends on the density of prime numbers among all 200 digit numbers

Density of primes: a first step • Theorem: there an infinite number of prime numbers. – Euclid ~300 B. C.

The Prime Number Theorem •

The Prime Number Theorem • To get a 200 digit prime number, we need the density of primes in the range from 10199 to 10200. ln(x) ~ 2. 3·(# digits in x)