The Distribution of the Primes
The distribution of both the composite and the prime numbers in our number system is the essential outcome of a number system constructed of repeated cycles of a sequence of numbers, in our case in decades, that is, as the numbers 1 to 10. The determination of the distribution of the composite numbers is more fundamental than the determination of the distribution of the primes, the primes being those numbers that remain “unchecked” when all the composite numbers in a sequence (whether limited or unlimited) have been determined. The primes are, therefore, essentially negative instances – they are numbers that are not composite.
It will be demonstrated here that the composite numbers in our number system, and in any number system likewise characterized by repeated number sequences, occur lawfully (i.e. they can be predicted) in recurrent cycles whose length is determined by the repetitive character of that system. I will demonstrate this with regard to our number system, which is based on repeated decades, and when this is understood it will be apparent that exactly the same lawful characteristics will apply to any number system built on repeated number sequences. Once the composite numbers have been identified, the primes automatically emerge as, as I have said, negative instances.
From the standpoint that composite numbers are created by multiplication rather than characterized by divisibility, if we proceed up the number series in an orderly fashion each number in turn can multiply each other number in the total series (ad infinitum), and each resultant product will be a composite number with its place in the total series. Thus 2 can multiply 1, 2, 3, 4 etc, creating the composite numbers , 4, 6, 8 etc, which we know as the even numbers. We also see at once that the composite numbers produced by multiplying by a particular integer can never be consecutive numbers (since multiplication or division by 1 is ruled out by definition).
Similarly 3 can multiply 1, 2, 3, 4 etc to create the composite numbers , 6, 9, 12 etc. Some of these numbers are, as it were, redundantly composite as multiples of 3, as they are already composite as products of 2, but 9 is a new addition, divisible by 3 but not by 2. Multiplying by 4, itself a composite number, creates no new composite numbers. As another even number, with 2 as a factor, it merely reproduces the work of multiplying by 2. But multiplying by 5 produces , 10, 15, 20 etc and so identifies another new composite number, 15.
The number 2 makes every second number, that is, the even numbers, a composite number. As half of the numbers of each decade are even numbers, being divisible by 2, therefore half of all numbers in the infinite series will necessarily be composite numbers. If we envisage the products, that is the composite numbers, throughout the total series that will be created by the multiplying action of the numbers of the first decade of the series, it is clear that every second number will have been made a composite number by 2, every third by 3, every fifth by 5, and every seventh by 7, while the composite numbers potentially created by 4, 6, 8, 9 and 10 have already been accounted for by 2 and 3.
Readability of the Composite Numbers
Our number system is entirely fortuitously built on repeated decades. This condition exercises quite specific, but also fortuitous, constraints on the patent recognizability of divisibility of numbers, that is, on the immediate readability of their character as either composite or prime numbers.
Only integers that divide evenly into 10 will produce composite numbers that always coincide with the final number in each successive counting decade. Because 2 divides evenly into 10, the composite numbers built on 2 occur always in the same position within a decade – at numbers ending in 2, 4, 6, 8 and 0. 5 is the only other number within the first decade that divides evenly into it, and it creates, like 2, a predictable and recognizable pattern of composite numbers throughout the number system. The number 5 makes every fifth number, that is, ending in 5 or 0, a composite number, that is, two per decade, although the 0-ending numbers have already been accounted for by 2.
The composite numbers built on all other numbers are created in cycles that do not coincide with the decades of our number system.
If we look at the role of multiplication by the numbers of the first decade on the numbers from 11 to 20, we find that the effect is that 12, 14, 15, 16, 18 and 20 are all composite in character, but that 11, 13, 17 and 19, missing these operations, can only be divided by themselves or 1, and are therefore what are called primes.
While five of the numbers in the first decade of the number system are primes, because of the increasing number of multipliers involved and hence an increased number of composite numbers, only four are primes in the second decade, 11-20: the number of composite numbers in the second decade has increased from five to six. The proportion of composite numbers will increase as we move up the number series and more successive numbers are included in the multiplicatory process, and in balance the proportion of primes will decrease. However, as in the progress from the first to the second decade, the creation of composite numbers will be increasingly redundant, as a higher percentage of composite numbers will already have been created by multiplication on numbers further back in the number system – not just by 2, 3, 5, and 7, but by 11, 13, 17, and so on. To have any effect, the new numbers must, of course, be primes, which have not already had their composite numbers created for them by lower multipliers.
The Pattern of Composite Numbers
As pointed out earlier, because the early primes 2 and 5 are factors of 10, which is the repeating base of our number system, we can recognize the composite numbers of which they are factors by inspection – numbers ending in 2, 4, 5, 6, 8, or 0 (in an even number) or in 5 – in regular cycles of 10. Thus the majority of the composite numbers are recognizable by inspection. The action of 2 and 5 also determines that all primes beyond the first decade can only end in 1, 3, 7, or 9. Numbers ending in these digits may be primes, but they may also be composite numbers that are the products of numbers other than 2 and 5. Because all numbers other than 2 and 5, whether smaller or greater than 10, are not factors of 10, their character as composite number or not cannot be simply read off the number sequence in terms of their final digit, as is the case with the composite numbers built on 2 and 5.
However, like 2 and 5, each other number in the “total series” determines a regular pattern of composite numbers as it multiplies first itself, and then each number above it. Thus the remaining primes of the first decade – 3 and 7 – although not factors of 10, like 2 and 5 subtend recurring patterns of composite numbers, but unlike those of 2 and 5, these extend beyond the first decade, and, cycling in a manner determined by the decade structure of the number system, also, eventually, return to scratch at the start of a decade. But because they do not divide evenly into 10, they will create differing numbers of composite numbers per decade, and with higher primes, in many decades none at all. However, there is regularity in their creation of composite numbers, as follows:
The rate at which each successive [prime] number creates composite numbers (redundant or otherwise) is 10 per 10n. That is:
2 creates 10 composite numbers in each 2 decades
3 creates 10 composite numbers in each 3 decades
5 creates 10 composite numbers in each 5 decades
7 creates 10 composite numbers in each 7 decades
The composite numbers built on each prime form a cyclical pattern determined by the pace at which one of its products first, and regularly thereafter, coincides with the final term of a standard decade: that is, when a product it creates ends in 0. This only occurs when 10n is reached. Each endlessly repeats this pattern, recommencing at the beginning of the decade after each cycle of 30, 70, 110 etc. The size of these cycles increases as the number series is ascended, and thus composite numbers are added at an ever slower pace.
If we look at the first 50 numbers in the number series, and identify the composite numbers created by 2 only, we find there are 19 (this leaves 31 numbers as potential primes); if ditto by 3, we find 15 (35 remain); if ditto by 5, we find 9 (41 remain); and if ditto by 7, we find 6 (44 remain). However, because some of these composite numbers are the creation of more than one of these four multipliers, the total number of composite numbers is less than the sum of their creations (that is, 49), and is in fact only 34, leaving a potential 16 prime numbers. This is, in fact, the total number of primes in the first 50, as the next prime to multiply itself is 11, whose product, 121, is outside this range. Nor is a greater number of composite numbers added in the next 50 by the primes greater than 7, by 11, 13, 17, 19 etc, because their squares moves into the range 100-400. The prime number 23 will not contribute a new composite number until we encounter its square, 529, and the prime number 37’s first novel composite number is 1369. The products of their multiplication by composite numbers and primes earlier in the number series than themselves are, of course, already registered as composite numbers in the series.
The large majority of composite numbers continue to be contributed by the primes of the first decade of the number system, by 2, 3, 5 and 7, at roughly a rate of 38-39 per 50 numbers after getting into full gear in the third decade, leaving a potential of 22-24 primes per 100, which will be successively reduced as composite numbers based on the succeeding primes – 11, 13 etc – invade their ranks with ever more sparse regularity.
Distribution of the primes
It can be seen that the distribution of the composite numbers, as expounded above, determines the distribution of the primes by omission, and explains the observed characteristics of their distribution.
The increasing entry of new cycles of composite number creation into the system means that the incidence of primes decreases as the number series advances. However, the increasing length of cycle for each successive composite number cycle that enters the system means that the rate at which the incidence of primes diminishes decelerates as the series advances. An additional factor is that the level of redundancy of the ten new composite numbers per cycle must increase as more cycles enter the system.
A further significant feature of the pattern of composite numbers which is integral to the structure of our decade-based number system is that the eight potential primes (i.e. non-composite numbers) per 30 numbers remaining after the composite numbers based on 2, 3 and 5 are determined (which together account for a large proportion of composite numbers in the system), fall into two clusters in the advancing number series, within each of which potential primes are within 1 to 3 positions of each other, while the two clusters are separated by gaps of 5 numbers. This variation in opportunity underlies the clustering of primes that occurs from time to time throughout the number series, despite their diminishing incidence.
If there were to be a highest prime number, it would occur when sufficient cycles had entered the system to entirely choke out the residual of the potential 8 primes per cycle of 30 when the cycles of 2, 3 and 5 have been accounted for. Whether this can occur (and proofs suggest that it does not) depends on the rate at which composite numbers are generated as against the rate at which new prime numbers occur in the advancing number series.
Generalizability to other Number Systems built on repeated Number Sequences
This understanding of the mechanism of the distribution of the composite numbers in our number system, which is built on the repetition of sequences of 10, is applicable to any number system built on repeated sequences. For example, for a number system built on repeated sequences of 7 digits, the repeating cycle for each number in turn would be that number multiplied by 7 (instead of by 10), so that for 3 in that system there would be a repeating cycle of 21 (though not of course written as 21, but as 30), and 5 in that system would have a repeating cycle of 35 (written as 50); or in a number system based on 12, the cycle for 3 would be 36 (3×12) numbers long (though written as 30) and for 5 it would be 60 (5×12) numbers long (written as 50). The distribution of composite numbers in any such number system would be documentable by the same processes as I have described for the decimal number system – some easily by inspection but others more laboriously by the process of successive multiplication, and with different parameters depending on the particular number base of the system – but in each case resulting in the progressive “sieving out” of the primes.
In each system a certain stable number of composite numbers would be created in the cycle based on each prime number in turn, although the number would differ depending on the sequence base – i.e. it would not in every case be 10. Further, one cannot expect the same presence of “readable” composites in all number systems. In number systems built on odd numbers such as 5 and 7, in the example above, there would be no readable composites in that no number within the repeated sequence would divide evenly into the basic span (as 2 and 5 divide into 10).
Why describe the Distribution of the Primes?
In that the distribution of the composite numbers, rather than the distribution of the primes, is the primary phenomenon, with the primes emerging by default, is it not, perhaps, perverse to concentrate on this remainder, rather than on the essential multiplicatory patterning in the creation of composite numbers, when describing this feature of number systems? Indeed, some number systems may have no primes.