34 1. Elementary Prime Number Theory, I

Exercises

1. (Harris [Har56]) Let b0,b1,b2 be positive integers with b0 coprime to b2.

Define Ak for k = 0, 1 and 2 as the numerator when the finite continued

fraction

b0 +

1

b1 +

1

...

+

1

bk

is put in lowest terms. For k = 3, 4,... , inductively define bk and Ak by

bk = A0A1 · · · Ak−3

and Ak by the rule given above. Prove that the Ai form an increasing

sequence of pairwise coprime positive integers.

2. (Aldaz & Bravo [AB03]) Let pi denote the ith prime. Euclid’s argument

shows that for each r, there is a prime in the interval (pr,

r

1

pi + 1].

Prove that the number of primes in the (smaller) interval (pr,

r

2

pi +1]

tends to infinity with r. Suggestion: With P =

r

2

pi, show that P −

2,P −

22,...,P

−

2k

are 1 and pairwise coprime for fixed k and large

r; then choose a prime factor of each.

3. (Chowdhury [Cho89]) It is trivial that for n ≥ 1, the number n!+1 has

a prime divisor exceeding n. Show that for n ≥ 6, the same holds for

each of the numbers n! + k, where 2 ≤ k ≤ n.

4. (Hegyv´ ari [Heg93]) Suppose a1 a2∑ a3 . . . is an increasing se-

quence of natural numbers for which 1/ai diverges. Show that the

real number α := 0.a1a2a3 . . . formed by concatenating the decimal ex-

pansions of the ai is irrational. In particular, 0.235711131719 . . . is ir-

rational. Hint: First show that every finite sequence of decimal digits

appears in the expansion of α.

Remark. Suppose that in place of our divergence hypotheses, we as-

sume that for each fixed θ 1, the number of ai ≤ x exceeds

xθ

for all

suﬃciently large x. Then Copeland & Erd˝ os [CE46] have proved that

the number α constructed above is normal (in base 10); in other words,

not only does every finite digit string appear in the expansion of α, but

each string of length k appears with the expected frequency

10−k.