Week 1: First Examples Mathematics is the art of explanation. Paul Lockhart Squares and Primes There’s something a little strange about the “square” numbers (1, 4, 9, 16, 25, 36, etc.): the only time a square number comes right after a prime number is when the square number is 4, which follows the prime number 3. Test this out with a few other square numbers, and you may begin to suspect, like I did, that a square number will rarely, if ever, come right after a prime number. What I am claiming, though, is that no square number aside from 4 will ever come immediately after a prime number. It can never happen. But what a broad generalization this is! How can I be so sure of this statement? After all, we’re talking about a set of numbers that go on forever (the square numbers) – and don’t the prime numbers also go on forever? Let me explain how I can make such a bold claim about these infinite families of numbers with so much confidence. Let’s say you pick some square number. Let’s call the number “푛,” just for the sake of having a name for it. What does it mean when I say 푛 is “square?” It means 푛 is the square of some other number, like 1, 2, 3, 4, etc. In other words, let’s say 푛 is the square of some positive whole number, which we might refer to as “푚” (or whatever letter you’d like to call it). So, in short, 푛 = 푚2. Now, what number comes right before 푛? That would be the number 푛 − 1, or in other words, 푚2 − 1. Why on Earth couldn’t 푚2 − 1 be a prime number? Think about what a prime number is. Can a number like 푚2 − 1 be written as the product of two other positive whole numbers? Remember that we can “factor” 푚2 − 1, like this: 푚2 − 1 = (푚 − 1)(푚 + 1). Now, 푚 − 1 and 푚 + 1 are both whole numbers, so no matter what 푚 is, 푚2 − 1 can always be factored into two whole numbers (specifically, whatever the numbers 푚 − 1 and 푚 + 1 are). Since the number 푚2 − 1 can always be written as the product of the numbers 푚 − 1 and 푚 + 1, does that mean 푚2 − 1 is never prime? Almost; the one and only way 푚2 − 1 could be prime is if one of those two numbers (either 푚 − 1 or 푚 + 1) turns out to be 1. Since 푚 is a positive whole number, 푚 + 1 has to be at least 2. But can 푚 − 1 be equal to 1? Sure it can! But only if 푚 is 2. In short, the only way 푚2 − 1 can be prime is if m is 2. In other words, the only way a prime number could come right before a square number is if that square number is 4. Week 1 Page 2 The previous three paragraphs constitute an informal proof, in conversational English, of the statement I made earlier: Aside from 4, no other square number can immediately follow a prime number. Isn’t it astounding and beautiful to realize that human beings can know something so infinite in scope with such absolute conviction? This kind of infinite certainty can only come from a mathematical proof. In mathematics, a proof is an airtight explanation of why a statement must be true, leaving absolutely no room for doubt. Exercise 1: Aside from 4, no other square number can immediately follow a prime number. Can a similar statement be made about the cubic numbers (1, 8, 27, 64, etc.)? Write the statement and prove it. Exercise 2: Can a similar statement be made about the fourth-power numbers (1, 16, 81, 256, etc.)? Write the statement and prove it. More Concise Proofs A mathematical statement that we intend to prove is often called a proposition. Let’s take another look at the following proposition and its proof: Aside from 4, no other square number can immediately follow a prime number. You will find that when you want to prove a proposition, it will be easiest to compose the proof if the proposition can be rewritten in the form “If , then .” For example, the proposition above can be rewritten as follows: If a square number immediately follows a prime number, then the square number must be 4. A more concise version of the proposition is this: If 푚2 − 1 is prime (푚 being a positive whole number), then 푚 = 2. Note that the three versions of the proposition given above are all stating exactly the same mathematical fact. Most mathematical propositions can be written in the form “If , then .” The two blanks here represent simple propositions like “푚2 − 1 is prime (푚 being a positive whole number)” and “푚 = 2.” For convenience, we can refer to these two propositions as “푃1” and “푃2.” The most direct way to try to prove a proposition in the form “If 푃1, then 푃2” is to suppose that the proposition 푃1 is true, and then explain why the proposition 푃2 must be true. This is what is done in the proof on the previous page, in a very conversational style. Below is a more concise style of writing the same proof. Proposition: If 푚2 − 1 is prime (푚 being a positive whole number), then 푚 = 2. Week 1 Page 3 Proof: Suppose 푚2 − 1 is prime and 푚 is a positive whole number. Since 푚2 − 1 = (푚 − 1)(푚 + 1), and 푚 + 1 cannot be 1, we know that 푚 − 1 = 1, and therefore 푚 = 2, which completes the proof. The first mathematical proofs, written by the ancient Greeks, ended with the phrase “quod erat demonstrandum” which can be roughly translated as “which completes the proof.” Until recently, all mathematical proofs (even in English) would end with the abbreviation “Q.E.D.” If you read a proof in a more modern high school or college mathematics textbook, you may see some special marking to visually indicate the end of the proof. The symbol often has a square shape, which is sometimes referred to as a “tombstone” mark. In my writing, I will use the “∎” symbol1 to indicate that a proof has concluded. Even Squares Think about the square numbers again. As it turns out, if a square number is even, then its “root” must also be even. In other words, if the square of a positive whole number 푛 is even, then 푛 must be even. This proposition has the form “If 푃1, then 푃2,” so you might feel the urge to prove the proposition in roughly the same way as the previous one: suppose the square of a positive whole number 푛 is even, and then explain somehow why 푛 itself must be even. What would it mean to say that 푛2 is even? To say that 푛2 is “even” means that 푛2 is a multiple of 2. That is, 푛2 = 2푚 for some positive whole number 푚. How would we use this to explain that 푛 itself must be a multiple of 2? The answer is that there is no direct way to explain this. We will take a different approach. Notice that the following two statements mean exactly the same thing: 1. If the square of a positive whole number 푛 is even, then 푛 must be even. 2. If a positive whole number 푛 is not even, then 푛2 cannot be even. Can you see that these two sentences are just two different ways of saying the same thing about positive whole numbers? The sentences mean the same thing for the same reason that the following non-mathematical sentences do: If it is raining, then the Smiths stay indoors. If the Smiths don’t stay indoors, then it isn’t raining. The fact that the two mathematical propositions above mean the same thing does not necessarily guarantee that either statement is true. But as it turns out, the second proposition is not terribly hard to prove, and if we prove it, the first proposition must also be true. 1 The “∎” symbol can be created in Microsoft Word by typing “220e” and then immediately pressing ALT+X. Week 1 Page 4 Now, thinking about proving proposition #2 above raises a new question – what does it mean for a positive whole number to be “not even”? Most people would say that a positive whole number that is not even is “odd,” but that is not very helpful. “Odd” is just a synonym for “not even.” Here is a more convenient definition: a number that is 1 less than an even number is called “odd.” Proposition: If the square of a positive whole number 푛 is even, then 푛 must be even. Proof: We will prove the equivalent proposition that if a positive whole number 푛 is odd, then 푛2 must be odd. Suppose a positive whole number 푛 is odd. This means 푛 = 2푚 − 1 for some positive whole number 푚, and so 푛2 = (2푚 − 1)2 = 4푚2 − 4푚 + 1. We can rewrite this as 2(2푚2 − 2푚 + 1) − 1, which is 1 less than an even number. This proves that if a positive whole number 푛 is odd, then 푛2 must be odd. ∎ Exercise 3: Explain why the sentence “If it isn’t raining, then the Smiths don’t stay indoors” does not mean the same thing as the other two sentences about the Smiths given before. Exercise 4: Prove that if 푎 is even and 푏 is odd, then 푎 + 푏 is odd.