Observations on the Regularity of Prime Number Distribution
In Terms of Highly Divisible Integers
by
Peter Marteinson, Ph.D.
Co-Editor,
Applied Semiotics / Sémiotique appliquée (University of Toronto)
Submitted To:
BRUCE PALKA
American Mathematical Monthly
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF TEXAS AT AUSTIN
1 UNIVERSITY STATION C1200
AUSTIN, TX 78712-1082
31 January, 2004
Stanislaw Ulam’s (1964: 516) most general observation on his famous spiral, that a
“property of the visual brain” allows patterns relating to the characteristics of primes to be
discovered, may indeed stimulate the mathematical imagination, and inspire further creative
attempts at visual pattern recognition in this area, but his spiral, like its derivatives, has yet to be
successfully interpreted in terms of possible arithmetic principles that can explain the genesis of
the known distribution of prime numbers. A corollary of this somewhat disappointing observation
is that Euler’s pessimistic prognosis has yet to be disproved: “Mathematicians have tried in vain
to this day to discover some order in the sequence of prime numbers, and we have reason to
believe that it is a mystery into which the mind will never penetrate” (cited by Ivars Peterson in
Science News, 5/4/2002).
Ulam’s spiral does, however, corroborate the intuition that the entire question of
primeness, articulated on the basis of an arithmetic product of two factors, is inherently a
problematic in two dimensions, in that a pair of factors, along with their product, may be
understood as a length, a width and an area in discretely quantified whole units of twodimensional space. Consequently, by representing numbers as two-dimensional matrices of dots,
one quickly observes that primes may be thought of as particularly ‘rough’ or uneven in terms of
the simple property of rectangularity.
The primes 11 and 13 both display this property (see Figure 1): no matter how many
permutations one attempts, rearranging their component ‘dots’ into differing rows and columns
never arrives at a rectangular configuration, as there is always a single exclusion, a ‘missing’
element, or a single additional element, in one of the rows. Their common neighbor, in contrast,
can be arranged in numerous fashions, each of which displays perfect rectangularity: two rows of
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six, three rows of four, four rows of three, or six rows of two. (One might consider, in any case,
that an arrangement of the primes in a single row of either eleven or thirteen elements is no longer
necessarily two-dimensional, and that such an arrangement merely constitutes a spatial
representation of the “one allowed” factorization of each prime, p x 1.) Similar representations of
Figure 1: A pair of primes, and their
common neighbor, in two dimensions
other primes, whether or not paired as twin primes, show the same result: not only are all primes,
when so represented, irregular in any quasi-rectangular configuration, but in addition, each may
be seen as a variant of a highly regular adjacent integer, 1 having been added or subtracted, and
in so doing, its high degree of regularity having been completely disrupted.
One potentially fruitful avenue to investigate, in attempting to identify some arithmetical
significance in the visual display of primes in this two-dimensional manner, appears therefore to
be to examine the properties of those ‘highly rectangular’ numbers to which primes are adjacent.
Such a line of investigation in fact reveals several significant observations.
Of Prime Numbers and ‘Prim’ Numbers
First among these observations is that prime numbers cannot occur adjacent to just any
number having rectangular proportions in two dimensions: take 9 and 15, for example, which can
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be represented, respectively, as three rows of three and three rows of five. Neither of these two
integers, however rectangular in two dimensions, has any prime numbers as neighbors (8 and 10
are composite, as are 14 and 16). An exhaustive investigation of such possibilities between 1 and
500 suggests that only certain regular, rectangular numbers, which I call ‘prim’ numbers because
they are ‘prim and proper,’ lead to prime numbers simply by the addition or subtraction of one.
These may be loosely defined as those composite integers having the highest number of distinct
factorizations in their immediate neighborhood of, say, n±3. In other words, a graph of n vs the
number of its factor pairs (other than n x 1) reveals an undulating relation, on which there are
regular maxima: these maxima are the ‘prim’ numbers: those divisible by a high proportion, or
nearly all, inferior non-prime integers (see Figure 2 below).
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
0
0
1
0
1
0
1
1
1
0
2
0
1
1
2
0
2
0
2
1
1
0
3
1
1
1
2
0
3
0
2
2x2
3x2
4x2
3x3
5x2
6x2
4x3
7x2
5x3
8x2
4x4
9x2
6x3
10x2
5x4
7x3
11x2
12x2
8x3
6x4
5x5
13x2
9x3
14x2
15x2
16x2
7x4
10x3
6x5
8x4
Figure 2: n and its factors from 1 to 32
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This exercise has been carried out for the first several hundred integers (see appendix). Yet
even in so brief an investigation, it becomes clear, furthermore, that ‘prim’ numbers occur at every
multiple of six (represented as yellow), and more interestingly, (pink) primes only occur at nprim
±1, or one integer from prim numbers. I hypothesize that this rule applies for all primes. However,
as no proof of this conjecture is yet available, let us be satisfied for now with the following
thought experiment as evidence in its support.
A ‘Displacement Principle’
Why is it that integers having a maximum number of factors are immediately adjacent to
integers with a minimum of whole factors? Clearly, if a prim number nprim 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 nprim+f. Similarly, the nearest integer inferior to nprim that is evenly divisible by f must
occur at nprim-1. This means that adding or subtracting an integer less than f is never sufficient to
reach a new integer that is also divisible by f. Therefore, a hypothetical prim number nprim having
as its factors all or nearly all imaginable inferior non-prime integers as factors, must be adjacent
to a pair of numbers, nprim+1 and nprim-1, that have no factors, or nearly none. Thus it is not
surprising that the most highly divisible integers, the multiples of 6 (which incidentally make up
the Assyrian, Sumerian and Babylonian systems for measuring time and angles that we inherited
in the West) are adjacent to each and every occurrence of prime numbers. Stated otherwise, all
integers adjacent to multiples of 6 have zero factors and are therefore prime, unless by
‘coincidence’ they themselves have a single prime pair by which they are divisible. No other
possibilities are observed. All numbers in this particular nprim±1 position either have zero or one
pair of factors other than one and themselves.
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A Modulus-6 Clock Spiral
The conjecture appears to be further strengthened by what I propose as a possibly clearer
alternative to Ulam’s checkerboard-type square spiral: a clock-like spiral in which the integers are
positioned according to a modulus-6 pattern (see Figure 3).
Figure 3: The Primes to 60 among Integers Revolving Modulus-6
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In which cases are the candidates for primeness suggested by this model, those integers
either 1 less or 1 greater than multiples of six, not prime? Again, the answer, which appears to lead
to a simple new algorithm for the computation of primes, is simple: if such a candidate, nprim±1,
is not prime, then it is composite, and can therefore be written as the product of two primes (or
one prime and one composite). 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 new kind of sieve, those prime numbers less
than or equal to a candidates’ square root. If none of these divides evenly, there are no factors
(other than one) and the candidate is indeed prime.
A Simple New Algorithm
To find all primes up to n one therefore, if this conjecture is correct, need only:
a) stop at each multiple of 6, m
b) consider m+1 and m-1 as candidates
c) test each candidate by dividing by every prime number # %m
d) conclude a candidate is prime if each such test in c) shows non-integer dividends.
The implications of this conjecture may be quite far-reaching in the field of cryptography:
if such an algorithm is correct, it may reduce the number of calculations required to factor large
primes by a factor of six or more. One possible consequence of this might be that soon the
information technology industry may choose to move to a higher standard for data encryption.
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Appendix:
File: prim e-analysis.xls
Works Cited
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.
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