Here are some interesting facts about natural numbers. I was about to call them “trivia,”
but after thinking about it for awhile, I decided that the ones involving primes are not trivial at all!
Divisibility
If you want to know whether or not a large number can be divided by a single-digit number without leaving a remainder, there are some handy little tricks you can use. You can use a cal-culator to see immediately whether or not any number is “cleanly” divisible by any other, but the following rules can be interesting anyway.
• A natural number is divisible by 2 without a remainder if it is even.
• A natural number is divisible by 3 without a remainder if the sum of the digits in its numeral is a natural-number multiple of 3.
• A natural number is divisible by 5 without a remainder if its numeral ends in either 0 or 5.
• A natural number is divisible by 9 without a remainder if the sum of the digits in its numeral is a natural-number multiple of 9.
• A natural number is divisible by 10 without a remainder if its numeral ends in 0.
You can combine these tricks and get the following facts:
• A natural number is divisible by 4 without a remainder you get an even number after dividing it by 2.
• A natural number is divisible by 6 without a remainder if it is even and the sum of its digits is a natural-number multiple of 3.
• A natural number is divisible by 8 without a remainder if you can divide it by 2 and get an even number, and then divide that number by 2 again and get an even number.
There aren’t any convenient tricks, other than using a calculator or performing “long divi-sion,” to find out if a natural number is divisible by 7 without leaving a remainder.
Is there a largest prime?
Now that you know what a prime number is, and you know that any nonprime natural num-ber can be broken down into a product of primes, you might ask, “Is there a largest prime?”
The answer is “No.” Here’s why. You might have to read the following explanation two or three times to completely understand it. Try to follow it step-by-step. If you can accept each step of this argument one at a time, that’s good enough. The fact that there is no such thing as a largest prime is one of the most important facts, or theorems, that have ever been proven in mathematics.
Let’s start by imagining that there actually is a largest prime number. Then we’ll prove that this assumption cannot be true by “painting ourselves into a corner” where we end up with something ridiculous. Now that we have decided there is a largest prime, suppose we give it a name. How about p? Theoretically, we can list the entire set of prime numbers (call it P).
It might take mountains of paper and centuries of time, but if there is a largest prime, we can eventually write all of the primes. We can describe the set P in shorthand like this:
P= {2, 3, 5, 7, 11, 13, ..., p}
Suppose that we multiply all of these primes together. We get a composite number, because it is a product of primes. No doubt, this number is huge—larger than any calculator can Natural Number Nontrivia 43
44 Natural Numbers and Integers
display—but it will be finite. Let’s call it y. What if we add 1 to y, getting a number even larger than the product of all the primes? If you call that new number z, you can express it like this:
z=y+ 1
= (2 × 3 × 5 × 7 × 11 × 13 × ... ×p)+ 1
Now we know that z has to be larger than p, because z is 1 more than, say, 2 ×p or 3 ×p or 5 ×p or 7 ×p. But there’s something else interesting about z. If we divide z by any prime number, we always get a remainder of 1. That’s because if we divide y by any prime, there’s no remainder, and z is exactly 1 more than y.
We know that z can’t be prime, because we’ve already determined that z is bigger than p, and we have already assumed that p is the largest prime. So z is composite. Because z is composite, it must be divisible without a remainder by at least one prime, that is, one element of set P. But wait! We just figured out a minute ago that if we divide z by any element of P, we get a remainder of 1. Therefore, z can’t be composite. But it can’t be prime either. But every natural number larger than 1 is either prime or composite! But ...
but ... but ... we are trapped!
There’s only one way out of this situation. Our original assumption, that there is a largest prime number, must be false. Reductio ad absurdum, which we first encountered in the solu-tion to Prob. 5 at the end of Chap. 2, comes to the rescue again!
How many primes?
The discovery that there is no largest prime number leads us straightaway into another impor-tant truth: there are infinitely many prime numbers. When a mathematician proves a major theorem like the one we just explored, and then some other fact follows on its heels, that secondary fact is called a corollary.
Let’s start out by assuming that the number of primes is finite, and load up our reductio ad absurdum “cannon” for another shot. This time it’s going to be easy. If the number of primes is finite, we can list them all. That means one of them has to be larger than all the others, so it is the largest prime. But we just discovered that there is no largest prime. Contradiction! The number of primes can’t be finite, so it must be infinite.
Are you confused?
When you have found the prime factors for a composite number, you can write the product out in any order. But unlike a listing of the elements of a set, in which you are allowed to list any element only once, you must be sure to include all the occurrences of a prime factor if it appears more than once. Take this example:
6,615= 3 × 3 × 3 × 5 × 7 × 7
You will get into trouble if you say, “The set of prime factors of 6,615 is {3, 5, 7}.” How do you know whether a given factor occurs once, twice, three times, or more?
Some people get around this issue by putting a little superscript called an exponent after a number in the set to indicate that it occurs more than once as a factor. They write that the set of prime factors of 6,615 is
{33, 5, 72}. That’s okay if you can remember that 33 does not literally mean 27 in this context, and 72 does not literally mean 49. (Neither 27 nor 49 are prime!)
The clearest way to express the prime factors of a composite number is to write out the product, list-ing each factor as many times as it “deserves,” and uslist-ing multiplication symbols between them. You can arrange the product in any order, but it helps if you start with the smallest factor and go up, or start with the largest factor and go down.
There is only one way to factor a composite number into a product of primes. This fact is called the Fundamental Theorem of Arithmetic.
Here’s a challenge!
The numbers 2 and 3 are both prime, and they are also consecutive whole numbers. Are there any other examples of two consecutive whole numbers that are both prime?
Solution
No, none of the even numbers larger than 2 (i.e., 4, 6, 8, 10, etc.) are prime. We can factor 2 out of any such number, and we always get a natural number bigger than 1. By elimination, then, all the primes larger than 2 are odd. If we take any of these primes and then find the next natural number, we are adding 1 to an odd number. That always produces an even number, which we have just seen can’t be prime.
The conclusion: when we come across any prime number larger than 2, the next consecutive natural number is always composite.