## Observations on the Regularity of Prime Number Distribution## Peter Marteinson |

Science News, 5/4/2002). The Encyclopædia Britannica (2004)
continues to suggest this consensus is alive and well, referring in an article entitled “Elementary Number Theory” to “the irregularity
in the distribution of primes” which “suggests that there is no simple formula for producing all the primes.”Figure 1. Ulam's Spiral Figure 2. Two primes and their common neighbor in two dimensions Of Prime Numbers and ‘Prim’ Numbers one integer from prim numbers.
The data imply that this rule must apply for all primes. ^{1} However, without yet attempting a proof, let us look
for confirmation through the following thought experiment as evidence in its support.A ‘Displacement Principle’ _{prim} is divisible by an integer factor greater than
one, f, then the next occurrence of an integer that is also evenly divisible by f occurs at n_{prim} +f. Similarly, the nearest integer
inferior to n that is evenly divisible by f must prim occur at n_{prim} -f. This means that adding or subtracting an integer less than f
is never sufficient to prim reach a new integer that is also divisible by f. Therefore, a hypothetical prim number n having as its
factors all or nearly all imaginable inferior integers as factors, must be adjacent to a pair of numbers, n_{prim} +1 and n_{prim} -1, that
have no factors, or a greatly reduced number of factors for that order of magnitude. Thus it is not surprising that the most highly divisible integers, the multiples of 6 (which incidentally
make up the Assyrian and Babylonian systems for measuring time and angles that we inherited in the West) are adjacent to each
and every occurrence of prime numbers greater than three. Stated otherwise, all integers adjacent to prims like multiples of 6 have zero factors and are
therefore prime, unless by mechanical ‘providence’ they themselves have a prime by which they are divisible. All numbers in this particular
n±1 position either have zero or very few non-trivial factors. Furthermore, this explains the role of the natural logarithm in the distribution of primes:
Looking at the appendix, an extended version of figure three, illustrates how factors are distributed in the progression of natural numbers — first every second natural number is given
a factor of two, and every third number from nine on is also given a factor, every fourth from sixteen on, etc., so we see that the probability
of having factors and being composite increases precisely according to the terms of the factorials that make up e and the very basis of the 1/log(n) frequency of prime distribution at n. These are 1 + 1/1 + 1/1x2 + 1/1x2x3 + 1/1x2x3x4 and so on.A Modulus-6 Clock Spiral |

Figure 4. A modulus-6 clock spiral showing the primes (red) to 90

_{prim} ±1, is not prime, then it is composite, and can therefore be written as the product of two primes. Furthermore, one of these two factors must be less than its square root, the other greater, unless they are equal
to each other and are the square root of that candidate. Therefore, one need only eliminate as factors, in a variant kind of sieve, those
prime numbers and near-primes less than or equal to a candidates’ square root. If none of these divides evenly, there are no factors
(other than 1) and the candidate is indeed prime.A Simpler Algorithm a) stop at each multiple of 6, mb) consider m+1 and m-1 as candidatesc) test each candidate by dividing it by each prime ≤√ md) conclude a candidate is prime if all such tests show a non-zero remainder. Conclusion p + q = 2n, where p and q are primes. It also illustrates graphically the manner in which
primes are created by the addition or subtraction of one from certain integers, which is the actual cause of the phenomenon that primes often occur in pairs. (This also implies that
3 and 5 are not paired primes in the same sense, though 5 and 7 are.)p,q mod6 = 5 must give a sum (10)mod6 = 4.e and the probability distribution 1/log(n).Notes1. Except 2 and 3, which clearly belong to a different category of primes on their own. The primes greater than 3, which are infinite in number, all occur adjacent to multiples of six. Also regarding prim numbers, any multiples of numbers themselves composed of many common small factors are also ‘very prim.’ For instance, multiples of both 3 and 4, meaning multiples of 12, are very prim, as are multiples of both 4 and 5, or of 20. Such numbers often occur on the ‘other side’ of a prime, with the multiple of six on the ‘first side.’ Further, although it has been observed in ancient Greek times that primes greater than 3 are all of the form 6n +/- 1, this appears not to have been explained by the highly divisible property of the multiples of six, and modular arithmetic, while frequently applied to the study of primes, does not appear to have been fully exploited to describe the orderliness of the primes, otherwise the Ulam sprial would likely not be a subject of such discussion in the literature (and nor would the Euler 41-400 square spiral). [Return 1] Appendix: Works Cited Collective (2004). Encylopædia Britannica, “Elementary Number Theory” (from “Number Theory”), Britannica, London. Peterson, Ivars (2002). “Prime Spirals,” Science News online, avail. http://www.sciencenews.org/20020504/mathtrek.asp Stein, M.L., S.M. Ulam, and M.B. Wells (1964). “A visual display of some properties of the distribution of primes.” American Mathematical Monthly 71(May):516-520. [Ver. 2.2, 8 June 05] E-mail to the editors Pour écrire à la rédaction © 2004, Applied Semiotics / Sémiotique appliquée |