A View of Primes, Twin Primes and Composites in Arithmetic Progressions

Iain Preston
2020 Figshare
The technology for using arithmetic progressions (AP) to study prime numbers (p) has been used extensively in the past and hence is not new. [1, 2, 3, 5, 6, 7, 19]. By applying the same technology I have chosen 8 related APs, each based on a common difference of 30 and each containing a set of unique numbers when compared to one another. As I am certainly not the first to utilise these particular 8 APs as a means of generating prime numbers [2], much of the paper will merely highlight why those
more » ... highlight why those 8AP are ideal for the exercise. For example, it will be highlighted how, by definition, they contain all possible prime numbers with the exceptions of the prime numbers 2, 3 and 5, and also how, when considered separately, they each contain an infinite number of primes. Whether or not it has also been considered previously, I will go on to show how, by classifying six of those 8 APs into three pairs, twin primes are to be found in each of those pairs, whereby one prime of a twin is found in one AP of a pair and the second prime of a twin is found in the other AP of the same pair. In doing so, I will then further demonstrate how the three pairs of APs when taken together contain all possible twin prime numbers with the exception of the twin primes 3 & 5 and 5 & 7, bearing in mind that each of those six APs will contain non-twin primes and composites too. Finally I will set out how the two remaining APs that are not paired, and hence not included in the six above, each contain an infinite number of primes none of which, apart from the prime number 7, can form one of a twin prime, and as such can be isolated from the other primes and twin primes. To the extent all 8 APs do each contain composites too, I will talk about the identification of composites and hence primes using sieve techniques, but since practical workable solutions using the same APs and computer programming techniques already exist [2], this is really "for information only".