How primesfit into the number jigsaw
How can we be sure that the primes do not become rarer and rarer and eventually peter out altogether?You might think that since there are infinitely many counting numbers and each can be broken down into a product of primes(something explained more carefully in a moment), there must then be infinitely many primes to do the job. Although this conclusion is true, it does not follow from the previous observations, for if we begin with afinite collection of primes, there is no end to the number of different numbers we can produce just using those given prime factors.Indeed, there are infinitely many different powers of any single prime:for example, the powers of the prime 2 are2,4,8,16,32,64,···. It is conceivable therefore that there are onlyfinitely many primes and every number is a product of powers of those primes. What is more, we have no way of producing an unending series of different primes the way we can,for example, produce any number of squares, or multiples of a specific given number. When it comes to primes, we still have to go out hunting for them, so how can we be sure they do not become extinct?
We will all be sure by the end of this chapter, butfirst I will draw your attention to one simple‘pattern’among the primes worth noting. Every prime number, apart from 2 and 3, lies one side or the other of a multiple of 6. In other words, any prime after these firsttwohastheform6n±1 for some number n.(Remember that6nis short for 6×n and the double symbol±means plus or minus.)The reason for this is readily explained. Every number can be written in exactly one of the six forms 6n,6n±1,6n±2, or6n+3 as no number is more than three places away from some multiple of six. For example,17=(6×3)-1,28=(6×5)-2,57=(6×9)+3;indeed, the six given forms appear in cyclic order,meaning that if you write down any six consecutive numbers, each of the forms will appear exactly once, after which they will reappear again and again, in the same order. It is evident that numbers of the forms 6nand 6n±2 are even, while any number of the form 6n+3 is divisible by 3. Therefore, with the obvious exceptions of 2 and 3, only numbers of the form 6n±1canbe prime.Thecasewhereboth of the numbers 6n±1areprime corresponds exactly to the twin primes:for example(6×18)±1gives the pair 107,109 mentioned in thefirst chapter. You might be tempted to conjecture that at least one of the two numbers6n±1 is always prime–this is certainly true for the list of primes up to 100 but thefirst failure is not far away:(6×20)-1=119=7×19, while(6×20)+1=121=112, so neither number is primewhenwetaken=20.