in our e-book Prime Passages to Paradise
by H. PeterAleff
Volume 1: Patterns of prime distribution
in "polygonal - number pyramids"
(To German translation - zur deutschen Übersetzung)
The six- wide array further helps to demonstrate the otherwise still unproven conjecture that there must be infinitely many twin primes, that is, pairs of numbers where p and p + 2 are both prime. Here is how:
In the six- wide rectangular array, the consecutive multiples of each number higher than three lay on a straight line from zero to that number and beyond, and on periodic parallels to that line further "down" if we begin writing the numbers from the "top" of the array. Soon after this "factor line" leaves the array rectangle on one side, a parallel to it re- enters it on the other side, farther down in the array at the next such multiple. Each so broken factor line thus cascades in evenly spaced stripes down the layers of the array.
(All the odd multiples of three are confined to the 6n + 3 column, so the factor line of three does not go through zero, and it will never interfere in the 6n ± 1 columns which alone can contain primes.)
Whenever the factor lines from all the primes above a given layer in that six- wide array happen to miss the two spaces before and after the 6n column in that layer, the entries there are not multiples of any among those prior primes. They are therefore primes themselves and form a pair of twin primes, as illustrated in Figure 3 and in the magnified detail below of the level on which those lines meet.
Now take the largest known pair of twin primes and imagine you plot the factor lines or stripes of all the smaller primes from five on.
As of late 2001, the largest known pair of twin primes has 33,220 digits, according to Chris K. Caldwell's Prime Pages. If you multiply all primes from five to that number with each other, their product becomes way, way greater.
The number of those smaller primes may be huge, but it is finite, so at some distance down the array, all their factor lines will meet on the same level.
They can meet either in a single odd space far down the array which represents the product of all said smaller primes, or they can meet much sooner, and quite often, in two or all of the three odd spaces on a single level in that six-wide array.
Actually plotting their factor lines on paper is of course out of the question, even if you micro- printed them on a roll of printer paper long enough to wrap the universe, but this is a thought experiment, and the size of the numbers involved does not matter because we can keep going down the array to infinity.
This approach corresponds to the way Euclid suggested to multiply all the primes, up to a supposedly "largest" one, with each other. He imagined this equally unfeasible multiplication to show that the result plus or minus one is either a prime, or else the product of two or more primes larger than the previously "largest". By this method, he proved that there always exists a prime larger than any allegedly "largest" one, and that there must thus be an infinite quantity of them.
Our meeting of the factor lines resembles the product of Euclid’s imaginary multiplication, except that we do not need to wait for all the factors to come together in one space but have also many earlier "distributed" meetings in the odd spaces of many intermediary layers. In each case, the candidate entries in the levels immediately before and after that meeting are not multiples of any of the primes united there because the factor lines of these are all too steep to reach the candidate spaces above or below the odd spaces of the level on which they meet.
Each entry at these next higher and next lower levels in the only two prime- containing columns of the six- wide array is therefore either a prime, or else the product of two or more primes larger than all those brought together at the meeting level.
The prime number theorem teaches that the density of primes decreases along a logarithmic curve as the numbers increase. The primes larger than the assumed "largest twin" are therefore more sparsely distributed than the smaller ones which unite their factor lines at that meeting level.
Those smaller primes make up the great majority of all composite numbers above that "largest" pair. Although the quantity of primes in the interval between the "largest twins" and the meeting of all smaller factor lines in a single space is higher than the number of primes up to those "largest twins", the larger primes contribute relatively little to the hail of multiples that cascades down the columns of the array and destroys the potential locations for primes.
Moreover, we do not have to wait for the meeting in a single space but encounter many earlier "distributed" meetings across intermediary layers.
The probability for a given space in the array to be hit by the factor line of a specific prime p is 1/p because it depends on how often that factor returns. Accordingly, the probability for that space to be hit by any prime corresponds the sum of the reciprocals of all the primes that are not otherwise excluded from hitting it, such as those that meet on an adjacent level.
That sum of prime reciprocals grows with sub- tectonic slowness. Although it will ultimately become as infinite as the number of primes itself if we keep going long enough, it takes a little more than the first 300,000 reciprocals to reach the sum of three, and the first million entries bring that total no higher than 3.068. The higher we go, the more that growth rate slows down further4a.
This means there is a low probability that one of those sparse factor lines from larger primes will hit one of the two candidate numbers in one of the two layers adjacent to a "meeting level".
The probability of two events occurring both is the product of their individual probabilities. The combined likelihood that both these meeting- adjacent levels would each catch one of those rare factor lines is therefore even lower than for a single hit.
However, that probability exists, so we repeat the process. We now bring together at a second group of meeting levels the factor lines of all the primes smaller than the numbers at the first meeting level. These second- tier meeting levels are again much farther down the array, but never mind, they still all follow obediently its rules.
The entries in the candidate spaces above and below each of those second- group meeting level are even less likely to encounter a factor line from one of the still sparser larger primes that occupy rows in the array farther down than said first meeting level.
Moreover, the combined likelihood of finding such a hit at each layer, in at least one of the two relevant spaces above and below each meeting level from the first and also second group, is again lower than the individual chances of hitting one of those spaces in a single layer.
As you repeat the process again and go to the third meeting level, and then to the fourth one, and so on far beyond any conceivable metaphor for the magnitudes of the numbers there, the probability of scoring such hits on the meeting- adjacent levels from the ever rarer extra factor trickle diminishes each time further. And the combined probability of consistently scoring all hits without exception diminishes even faster.
Continue repeating the process, and the combined probability for such an uninterrupted series of hits without ever missing a single one on each of the ever more successive candidate rows in the array -- that overall probability becomes soon vanishingly small. If you repeat the process infinitely often, this probability approaches zero.
The inverse probability, that the factor lines will miss at least one among all those rows, approaches one, or certitude. We can therefore be certain that the two candidate entries in at least one of the many directly pre- or post- meeting levels escape those ever sparser new factor lines and are thus both prime.
This means there is always a pair of twin primes larger than any allegedly "largest" one, and their succession has no end.
This is just an example of how useful such rectangular arrays can be for exploring the behavior of primes.
Residues from division by prime numbers have also helped to define so many other properties of primes that the mathematician Paulo Ribenboim, one of the score keepers in this field, followed his description of Eratosthenes’ sieve in his book on prime number records with a long section on fundamental theorems and primality tests that are all based on such remainders4.
However, the imposition of any constant array width upon the steadily growing sequence of numbers scrambles the salient visual patterns in the distribution of primes which more suitable methods of choosing the segment lengths reveal.
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