Most texts on Prime Numbers (and even a professor of Maths at Oxford University?) don’t seem to explain the patterns and order in the primes and instead prefer to use words such as “random”, “unruly or “mysterious”. There is no doubt their distribution has been difficult to explain but just because something is “complex” does not make it random or unexplainble . I hope you find the diagrams below useful in understanding prime distribution and for predicting where primes can occur and where they definitely won’t. Comments welcome as I am learning all the time.
Lines of Symmetry
Symmetry is not given much attention when considering prime number distribution. Hardly surprising as prime numbers don’t occur symmetrically! However if one looks at the positions that prime numbers occur it is clear that they occur only in certain locations, and don’t occur in other locations. In addition, the locations where primes do occur can be mapped using lines of symmetry. If you don’t believe me calculate p#/2 and then look at the differences between the primes and p#/2 either side of p#/2. Do that for each p#/2 yes and of course p#/2 gets big very quickly so it’s easy only with the smaller ones.
Of course if you have tried this exercise you will find all sorts of anomalies; primes that don’t seem to fit in but you have to persevere and the two diagrams below should assist you.
The first couple of lines of “symmetry” are well known. The first reflects the fact that most (not 2 or 3) prime numbers can be expressed in the form 6n+1 and 6n-1, but with the disappointing aside that not all such numbers are prime. Within each strand of primes they are distributed at intervals of 3# = 6. As with further lines of symmetry, the smallest prime numbers don’t fit in, but in a predictable way – the patterns are stepped, becoming more complex the larger the integer. Thus the symmetrical patterns do not all start at 2, but at increasing primes. This is fundamental to understanding how primes are distributed.
Thus, putting 2 and 3 to one side, the prime numbers can be considered to be either left handed with a formula of 5 + n.3# (5,11,17,23…) or right handed with a formula of 7 + n.3# (7,13,19,31….). In theory at least the line of symmetry starts at 3 (or 3#/2).
The next line of symmetry starts at 5#/2 = 15 AND the pattern it explains starts at a larger prime, 11. The primes occur at intervals of multiples of 5#=30 from 8 bases:
with ALL prime numbers distributed at equal intervals about the line 15-45-75-105…. provided that smaller primes are discarded first. This diagram reflects a number of features of the primes which are obvious in retrospect:
i) there are only three types of pair of prime numbers 1:3, 7:9 and 9:1.
ii) the 1:3 pairs match the locations of the 7:9 pairs about the line of symmetry; the 1:9 pairs are equidistant from the line of symmetry
iii) there are two lines of primes that never form pairs (the +8 and -8 lines), so we can speculate that prime pairs contain more primes that end in 1 and 9 and fewer that end in 3 and 7. ….that in some respects assumes that each of these 8 lines contain an equal frequency of primes….which is also an interesting thing to consider further.
iv) there are zones of 6 integers on either side of the line of symmetry in which primes never occur (i.e. between the +8 and +14 lines) between primes ending in _1 and _7 on the one hand and those ending between _3 and _9.
In consequence it can be conjectured that there are an infinite set of similar lines of symmetry in the prime numbers each starting at n#/2; thus
The lines of symmetry are not immediately obvious if one only has the set of prime numbers to start from. It is necessary to start from the Sieve of Eratosthenes in order to obtain the necessary groupings (to be described later).
The third line of symmetry starts at 105; primes occur at intervals of 7#=210. As shown it is not quite symmetrical and two lines from the bottom – starting 211 and 221 – have to be brought up to the top, but staggered, to provide the complete pattern. Some of these bases aren’t prime …..121, 143, 169, 187, 209, 221…but are predictable. How? Well they represent 11×11, 11×13, 13×13 (i.e.multiples of all primes 11 and greater). Notice again the pairing of prime pairs about the line of symmetry, and the absence of primes in some zones of length 6, and some areas with no pairs:
Bringing the last two lines up to the top gives the complete picture:
So looking at the pairs again:
There are 48 lines:
7 lines of 3 and 7 with no chance of pairing (14 lines)
2 lines of 1 and 9 with no pairs (4 lines)
So pairs only occur on 48 – 14 – 4 lines: 30 lines – 7 pairs of lines on each side of the line of symmetry and one pair that straddles – equidistant starting 209-211.