Download 5. A first look at primes Definition 1. An integer p > 1 is called prime if

5. A first look at primes
Definition 1. An integer p > 1 is called prime if its only positive divisors are 1 and p. An
integer m > 1 which is not prime is called composite.
The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23....
Exercise: Prove that if m is composite then m = ab for some a, b ∈ Z with 1 < a < m and
1 < b < m.
Primes are the “building blocks” of the integers, in the sense that every integer greater than
one is a product of primes. (In fact, this factorization is unique, an important fact we will show
later.)
Theorem 2. Every integer greater than one is a product of (one or more) primes.
Proof. Suppose this statement is false. Let n be the smallest integer greater than one which is
not the product of (one or more) primes. In particular, n is composite. By the Exercise, n = ab
for some 1 < a < n and 1 < b < n. Now, since n is the smallest integer greater than one which
is not the product of primes, a and b must both be products of primes. But then n = ab is the
product of primes as well. This is a contradiction. Consequently, the theorem must be true.
It is a famous theorem of Euclid (circa 300 BCE) that there are infinitely many primes:
Theorem 3. There are infinitely many primes numbers.
Proof. Suppose not. Then let {p1 , . . . , pn } denote the set of all primes. Consider the the integer
N = p1 p2 · · · pn + 1. Now, by the previous theorem, N must be a product of (one or more)
primes. In particular, some prime must divide N . Say pi | N . Then, since pi | p1 · · · pn , we must
have that pi | 1, a contradiction. Therefore, there are infinitely many primes.
Primes can be rather mysterious, and there are many questions about them which remain
unanswered. One of these is Goldbach’s Conjecture, which says that every even integer greater
than two is the sum of two primes. Another is the Twin Prime Conjecture, which says there are
infinitely many pairs of primes that differ by two (e.g., 5 and 7, 17 and 19, 41 and 43, etc.)
While there is no known pattern for the distribution of prime numbers, some primes have
special forms. A Mersenne prime is a prime number of the form 2n − 1. The first few Mersenne
primes are 3 = 22 − 1, 7 = 23 − 1, and 31 = 25 − 1.
Exercise: Prove that if 2n − 1 is prime then n must be prime. (Hint: Suppose n = ab where
1 < a < n and 1 < b < n. Then 2n − 1 = (2a )b − 1. Now use that xm − 1 = (x − 1)(xm−1 +
xm−2 + · · · + x + 1) for any x and m ≥ 1.)
Is it possible that 2p − 1 is prime for every prime p? The answer is no: 211 − 1 = (23)(89).
Currently, 47 Mersenne primes are known. It is not known whether there are infinitely many
Mersenne primes.
Another special type of prime number is a Fermat prime. These are prime numbers of the
form 2n + 1 for some integer n ≥ 1. The first few Fermat primes are 3 = 21 + 1, 5 = 22 + 1, and
17 = 24 + 1. In fact, on your homework you will show that if 2n + 1 is prime, then n must be a
power of 2. Again, one may ask whether 2n + 1 is always prime when n is a power of 2. (This
was one of Fermat’s famous conjectures.) Leonhard Euler in 1732 showed this is not the case:
232 + 1 = (641)(6700417).
Homework:
1. Prove that if 2n + 1 is prime and n ≥ 1, then n = 2m for some m ≥ 1. (Hint: By factoring
out all the two’s from n, we can write n = 2k b for some k ≥ 0 and where b is odd. Now
use that if b ≥ 1 is odd, then xb + 1 = (x + 1)(xb−1 − xb−2 + · · · − x + 1).)
2. Let n ≥ 1 and p1 , p2 , · · · , pn are the first n primes. Is p1 · · · pn + 1 always prime?
3. Let’s call three consecutive primes triplet primes if they are of the form p, p + 2, and p + 4,
for some prime p. Show that 3, 5, 7 are the only triplet primes.