17. Commensurability of segments. In hisElements, Euclid investigates alsocommon measuresof segments besides the common divisors of integers. A common mea- sure of two segments is a segment that can be measured an integer number of times onto both segments (without remainders). Two segments arecommensurableif they have a common measure.
(a) Prove that two segments are commensurable if and only if the ratio of their lengths is a rational number.
(b) How many common measures do two commensurable segments possess?
(c) Formulate the division algorithm for segments and show that the Euclidean algorithm terminates in finitely many steps if and only if the two original seg- ments are commensurable.
(d) Verify that commensurable segments have a greatest common measure and any common measure can be measured an integer number of times onto this greatest one (without remainder).
(e) Show that the side and the diagonal of a square are not commensurable (thus giving a geometric proof for the irrationality of√2).
1.4. Irreducible and Prime Numbers
We have seen that0and the units play special roles in divisibility: every integer divides 0and the units divide every integer. Consider now any integer𝑎different from0and units. By the definition of units,𝜀 ∣ 𝑎and𝜀𝑎 ∣ 𝑎 for every unit𝜀. These are called thetrivial divisorsof𝑎. The numbers having only trivial divisors are of distinguished importance:
Definition 1.4.1. An integer𝑝different from units (and zero) is calledirreducibleif it can be factored into the product of two integersonlyso that one of the factors is a unit:
𝑝 = 𝑎𝑏 ⟹ 𝑎or𝑏is a unit. ♣
We do not have to prescribe𝑝 ≠ 0because0has non-trivial factorizations too, e.g.0 = 5 ⋅ 0. We note further that in the product𝑝 = 𝑎𝑏, both factors cannot be units since then their product, i.e.𝑝, would be a unit as well. (Hence, the word “or” occurs at the end of Definition 1.4.1 in an “exclusive” sense.)
Thus, the irreducible numbers are those integers distinct from units that can be factored into the product of two integers only trivially, or otherwise stated, are divisible only by their associates and units. Such numbers are e.g.2,3,−17, etc. If a non-zero integer has a non-trivial divisor, then it is called acompositenumber.
Before introducing the following notion, we recall that if an integer𝑐divides a factor of a product, then𝑐necessarily divides also the product, but the converse is false:
e.g. for𝑐 = 6we have6 ∣ 3 ⋅ 4, but6 ∤ 3and6 ∤ 4. The numbers satisfying the converse are of special significance:
Definition 1.4.2. An integer𝑝different from units and zero is called aprimenumber (or shortly, just aprime) if it can divide the product of two integersonlyif it divides at
least one of the factors:
𝑝 ∣ 𝑎𝑏 ⟹ 𝑝 ∣ 𝑎or𝑝 ∣ 𝑏. ♣
At the end of Definition 1.4.2, the word “or” occurs in an inclusive sense since it can happen that𝑝divides both factors of the product. We also note that the restriction 𝑝 ≠ 0was necessary here since0 would otherwise satisfy the property required in Definition 1.4.2:
0 ∣ 𝑎𝑏 ⟹ 𝑎𝑏 = 0 ⟹ 𝑎 = 0or𝑏 = 0 ⟹ 0 ∣ 𝑎or0 ∣ 𝑏.
Definition 1.4.2 implies that if a prime divides a product of more (than two) factors, then it must divide at least one of them.
Theorem 1.4.3. Among the integers,𝑝is a prime if and only if it is irreducible. ♣ Proof. We may clearly assume that𝑝is not zero and not a unit.
I. First, we take a prime𝑝and prove that it is irreducible. Given a product𝑝 = 𝑎𝑏, we have to verify that𝑎or𝑏is a unit.
The equality𝑝 = 𝑎𝑏implies that𝑝 ∣ 𝑎𝑏. Since𝑝is a prime, therefore we infer that𝑝 ∣ 𝑎or𝑝 ∣ 𝑏. The first case means that𝑎𝑏 ∣ 𝑎and hence𝑏 ∣ 1(since𝑎 ≠ 0), i.e.𝑏is a unit. The second case yields similarly that𝑎is a unit.
II. We assume now that𝑝is irreducible and prove that it is a prime. Given𝑝 ∣ 𝑎𝑏, we have to verify that at least one of𝑝 ∣ 𝑎and𝑝 ∣ 𝑏holds.
If𝑝 ∣ 𝑎, then we are done. If𝑝 ∤ 𝑎, then the irreducibility of𝑝and(𝑝, 𝑎) ∣ 𝑝yield (𝑝, 𝑎) = 1. The conditions𝑝 ∣ 𝑎𝑏and(𝑝, 𝑎) = 1imply𝑝 ∣ 𝑏by Theorem 1.3.9. □ Thus we have shown that the irreducible and prime numbers coincide among the integers. Therefore we can define the prime numbers as in high school by the irre- ducible property and to use either of the two adjectives irreducible and prime for these numbers. For brevity, we shall generally use the word prime except if we want to em- phasize the irreducible property.
The two notions, however, are not equivalent in many other sets of numbers. E.g.
among the even numbers,6is irreducible since it cannot be written as the product of two even numbers, but it is not a prime because it divides18 ⋅ 2without dividing either of the factors. We shall see further examples in Chapter 10.
Among the integers, the study of prime numbers is one of the most important areas in number theory. Euclid proved that there exist infinitely many primes (Theo- rem 5.1.1), but on the other hand, there are many easily formulated and yet unsolved problems concerning the prime numbers. We shall deal with these more in detail in Chapter 5.
Exercises 1.4 23
Exercises 1.4
According to the conventions, we shall use the word prime or prime number also for the irreducible numbers among the integers. We note, however, that Exercises 1.4.1–
1.4.7 refer to irreducible numbers.
1. Determine all positive integers𝑛 for which each of the following numbers is a prime:
(a) 𝑛,𝑛 + 2, and𝑛 + 4 (b) 𝑛and𝑛2+ 8
(c) 𝑛,𝑛 + 6,𝑛 + 12,𝑛 + 18, and𝑛 + 24 (d) 𝑛,𝑛3− 6, and𝑛3+ 6.
2. Does there exist an infinite arithmetic progression with a non-zero difference con- sisting purely of primes?
3. Captain Immortal has three immortal grandchildren whose ages are three distinct primes and the sum of the squares of their ages is a prime. How old is the captain’s youngest grandchild? (Do not forget about the immortality of the grandchildren, they can be several million years old!)
4. Let𝑎and𝑘be integers greater than one. Prove the following assertions.
(a) If𝑎𝑘− 1is a prime, then𝑎 = 2and𝑘is a prime.
(b) If𝑎𝑘+ 1is a prime, then𝑘is a power of two.
Remark: The primes of the form2𝑘− 1are calledMersenne primesand the primes of the form2𝑘+ 1are calledFermat primes. We shall study them in detail in Sec- tion 5.2.
S 5. Determine all integers𝑡 > 1andoddnumbers𝑘 > 0for which1𝑘+2𝑘+3𝑘+⋯+𝑡𝑘 is a prime.
6. Find all positive integers𝑛for which (a) 𝑛3− 𝑛 + 3
(b) 𝑛3− 27
(c) 𝑛8+ 𝑛7+ 𝑛6+ 𝑛5+ 𝑛4+ 𝑛3+ 𝑛2+ 𝑛 + 1 (d) 𝑛4+ 4
(e) 𝑛8+ 𝑛6+ 𝑛4+ 𝑛2+ 1 is a prime.
7. Let𝑛 > 1. Prove the following assertions.
(a) If𝑛has no divisor𝑡satisfying1 < 𝑡 ≤ √𝑛, then𝑛is a prime.
(b) The smallest divisor of𝑛greater than1is a prime.
(c) If𝑛is composite but has no divisor𝑡satisfying1 < 𝑡 ≤ 3√𝑛, then𝑛is the product of two primes.
8. Prove that(𝑛 − 5)(𝑛 + 12) + 51is never divisible by289if𝑛is an integer.
9. Which will be the irreducible and prime elements among the even numbers?
10. The notion of irreducible and prime elements can be defined in any integral do- main𝐼(see Exercise 1.1.23). Prove the following propositions.
(a) If multiplication has no identity element in𝐼, then there are no primes in𝐼.
(b) If multiplication has an identity element in𝐼, then every prime is irreducible in𝐼.