Other proofs of the infinitude of the primes
n
1. (Goldbach, 1730). Let Fn = 22 + 1. So F0 = 3, F1 = 5, F2 = 17, F3 = 257, F4 = 65537, . . ..
(a) Prove (by induction) that
F0 · F1 · F2 · · · Fn−1 = Fn − 2.
(b) Prove that if i < j, gcd(Fi , Fj ) = 1.
(c) Let pi be the smallest prime divisor of Fi . Prove that p1 , p2 , . . ., are distinct.
2. (Euler, 1748).
(a) Fix a positive integer y and let N be the set of all integers all of whose prime factors are
≤ y. Use the FTA to show that
X
Y
1
1
1
1 + + 2 + ··· =
.
p p
n
p≤y
n∈N
(b) Show that
Ry
≥ 1 + 12 + · · · + y1 ≥ 1 1t dt = ln(y).
−1
.
+ p12 + · · · = 1 − p1
1
n∈N n
P
(c) Show that 1 +
1
p
2
(d) Show that for 0 ≤ v ≤ 1/2, ev+v ≥ (1 − v)−1 . Conclude that
Y
2
e1/p+1/p ≥ ln(y).
p≤y
P
P
P∞
1
1
(e) Conclude from (d) that
+ p12 ≥ ln(ln(y)). Show that
2 ≤
p≤y
p≤y
n=2
p
p
R∞ 1
dx
=
1.
2
1 x
P
P
(f) Conclude that p≤y p1 ≥ ln(ln(y)) − 1. Deduce that p prime p1 diverges.
1
n2
≤
3. (Erdős, 1938).
(a) A squarefree positive integer is one with the property that there is no prime p so that p2 |n.
Prove that every positive integer can be written uniquely as a product n = rs2 , where r is
squarefree and s ≥ 1.
(b) Assume there are finitely many primes k, say p1 , p2 , . . ., pk . Conclude that there are exactly
2k squarefree numbers, r1 , r2 , . . ., r2k .
1
2
(c) Since every positive integer can be written as n = ri s2 for some s, conclude that the number
of integers ≤ x is
r r r √
x
x
x
+
+ ··· +
≤ 2k x.
r1
r2
r2k
√
(d) Show that if x is large enough, x > 2k x and derive a contradiction.
4. (Furstenberg, 1955). If a ∈ Z and b ≥ 1 define Na,b = {x ∈ Z : x ≡ a (mod b)}. Define a
subset O ⊆ Z to be open if for any a ∈ O, there is some set Nc,d so that a ∈ Nc,d and Nc,d ⊆ O.
Define a set O to be closed if Oc = {x ∈ Z : x 6∈ O} is open.
(a) Show that a finite union of closed sets is closed.
(b) Show that any open set is infinite.
(c) Show that the set Na,b is closed, too.
(d) Assume there are finitely many primes p, say p1 , p2 , . . ., pk . Show that
k
[
N0,pi = {x ∈ Z : x 6= 1, x 6= −1}
i=0
is closed, and conclude that {1, −1} is open. Derive a contradiction.
5. (Saidak, 2005).
(a) Prove that if a and b are positive integers > 1 with gcd(a, b) = 1, then ab has more distinct
prime factors than either a or b.
(b) Let N1 = 2. Define Nk = Nk−1 (Nk−1 + 1). Use part (a) and induction to prove that Nk
has at least k prime factors.