Below we present solutions to the exercises. We do not include the computer assignments in these solutions. To theAdditional exercises not all solutions are fully worked; sometimes we only give hints. To facilitate the reading, we always repeat the exercise, sometimes in a condensed form, before giving the solution.
Chapter 1: Prime numbers
Exercise 1.1. a. Using table 1.1 find all primes smaller than100.
b. What numbers smaller than100 can you construct by multiplication using only the numbers 3 and 8 as building blocks?
c. Is it possible to construct 103 = 2 × 3 × 17 + 1 using only 2,3 and 17? What about 104?
d. Is the number2 × 3 × 5 × 7 × 11 + 1 prime? And what about 2 × 3 × 5 × 7 × 11 × 13 + 1?
a. There are 25 prime numbers less than 100, namely 2,3,5,7,11,13, 17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97.
b. 3, 8, 9, 24, 27, 64, 72, 81.
c. 103 is not divisible by one of the building blocks 2, 3 and 7, so it cannot be constructed. 104 is divisible by the building block 2, but 104 = 2 × 2 × 2 × 13 and the building block 13 (or 26, or 52, or 104) is missing. Both numbers thus cannot be constructed.
d. 2 × 3 × 5 × 7 × 11 + 1 = 2311 is a prime number, but 2 × 3 × 5 × 7 × 11 × 13 + 1 = 30031 = 59 × 509 is a composite number.
Exercise 1.2. a. Using the computer examine graphs of π (x) on various domains. First take 0 ≤ x ≤ 100, as in figure 1.1. This graph should also be in line with the outcome of exercise 1.1.a. Check this!
b. Also try the domains 0 ≤ x ≤ 1 000, 0 ≤ x≤ 10 000, …, 0 ≤ x ≤ 1 000 000.