Primes
Reading
The reading is contained in section 4.3. You should read all of it that you haven’t read already.
Summary
A integer p > 1 is prime if it has no positive divisors except 1 and itself. If an integer is not prime, it is composite.
Lemma 1. An integer p > 1 is prime if and only if it satisfies the following property:, for all integers a and b, if
p | ab then p | a or p | b.
Proof. Suppose p is prime. Then gcd(a, p) = p or gcd(a, p) = 1. If gcd(a, p) = p, we get that p | a and we are done.
Otherwise, a and p are relatively prime, so p | b, and again we are done.
Conversely, suppose p satisfies the given condition. If p were composite, we would be able to write p = ab for
some a, b ∈ Z where 1 < a, b < p. In particular, we have p | ab, so either p | a or p | b. But this is a contradiction,
since p cannot divide any positive integers smaller than it.
Theorem 2 (Fundamental theorem of arithmetic). Every integer n > 1 can be written uniquely as a product of
primes.
Proof. We’ll prove this later.
Theorem 3 (Euclid). There exist infinitely many primes.
Proof. Suppose there existed only finitely many primes p1 , . . . , pn . Consider theinteger a = p1 · · · pn + 1. By the
fundamental theorem, we know that a must have a prime factor. On the other hand, notice that the remainder
after dividing a by pi is 1 for all i. So none of the pi is a factor of a, which is a contradiction.
Proposition 4. Suppose a = pa1 1 · · · pann , and let b = pb11 · · · pbnn , where p1 , . . . , pn are primes and a1 , . . . , an , b1 , . . . , bn
are nonnegative integers. Then
min{a1 ,b1 }
n ,bn }
gcd(a, b) = p1
· · · pmin{a
n
and
max{a1 ,b1 }
lcm(a, b) = p1
Proof. Exercise.
1
n ,bn }
· · · pmax{a
.
n