Prime Numbers

50 Chapter 1 PRIMES!
prime? What is the largest integer N divisible by every integer smaller than
√ N?
1.2. Prove Euclid’s “first theorem”: The product of two integers is divisible
by a prime p if and only if one of them is divisible by p. Then show that
Theorem 1.1.1 follows as a corollary.
1.3. Show that a positive integer n is prime if and only if
∞
n
n − 1
− =2.
m m
m=1
1.4. Prove that for integer x ≥ 2,
x
1
π(x) = n k=2⌊⌊n/k⌋k/n⌋
.
n=2
1.5. Sometimes a prime-producing formula, even though computationally
inefficient, has actual pedagogical value. Prove the Gandhi formula for the
n-th prime:
⎢ ⎛
⎢
pn = ⎣1 − log ⎝− 2
1 µ(d)
+
2 2
d|pn−1!
d ⎞ ⎥
⎠⎦
.
− 1
One instructive way to proceed is to perform (symbolically) a sieve of
Eratosthenes (see Chapter 3) on the binary expansion 1 = (0.11111 ...)2.
1.6. By refining the method of proof for Theorem 1.1.2, one can achieve
lower bounds (albeit relatively weak ones) on the prime-counting function
π(x). To this end, consider the “primorial of p,” the number defined by
p# =
q =2· 3 ···p,
q≤p
where the product is taken over primes q. Deduce, along the lines of Euclid’s
proof, that the n-th prime pn satisfies
pn 1
ln ln x,
ln 2
for x ≥ 2.
Incidentally, the numerical study of primorial primes p# + 1 is interesting
in its own right. A modern example of a large primorial prime, discovered by
C. Caldwell in 1999, is 42209#+1, with more than eighteen thousand decimal
digits.