34 1. Elementary Prime Number Theory, I
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
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,
pi + 1].
Prove that the number of primes in the (smaller) interval (pr,
tends to infinity with r. Suggestion: With P =
pi, show that P −
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
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