How primes fit into the numberjigsaw
How can we be sure that the primes do notbecome 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 ofprimes(something explained more carefully in a moment), there must then be infinitely many primes to do thejob. Although this conclusion is true, it does not follow from the previous observations, for ifwe begin with a finite collection of primes, there is no end to the number ofdifferent numbers we can producejust using those given prime factors.Indeed, there are infinitely many different powers ofany single prime:for example, the powers of the prime 2 are2,4,8,16,32,64,…. It is conceivable therefore that there are only finitely many primes and every number is a product of powers ofthose primes. What is more, we have no way of producing an unending series ofdifferent primes the waywe can,for example, produce any number ofsquares, 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 ofthis chapter, but first 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 ofa multiple of 6. In other words, any prime after these first two has the form 6n±1 for some number n.(Remember that6n is 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 ofsix. 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 ifyou 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 6n and 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±1 can be prime. The case where both of the numbers 6n±1 are prime corresponds exactly to the twin primes:for example(6×18)±1gives the pair 107,109 mentioned in the first 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 ofprimes up to 100 but the first failure is not far away:(6×20)-1=119=7×19, while(6×20)+1=121=112, so neither number is prime when we take n=20.