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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 solution to Prob 5 at the end of Chap 2, comes to the rescue again!
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How many primes The discovery that there is no largest prime number leads us straightaway into another important 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
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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
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{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, listing each factor as many times as it deserves, and using 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
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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
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