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 (fig.1), 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. Of his spiral he says only that it “appears to exhibit a strongly nonrandom appearance” (Stein et al. 1964). 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). 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
AS/SA nº 14,
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 two-dimensional space.
Consequently, by representing numbers as two-dimensional matrices of elements, one quickly observes that primes may be thought
of as particularly ‘rough’ or uneven in terms of the simple property of rectangularity.
Figure 2. Two primes and their common neighbor in two dimensions
The primes 11 and 13
both display this property (see Figure 2): no matter how many permutations one attempts, rearranging their component elements
into differing rows and columns, one 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 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
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.
AS/SA nº 14,
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 integer, nor even adjacent to any number having rectangular proportions in two dimensions: take 9 and 15, for example, which can
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,’ or highly divisible, give prime numbers simply by the addition or subtraction of one. These may be loosely understood as those
composite integers having the highest number of distinct factorizations in their immediate neighborhood. Further investigation reveals
that the prim or most highly divisible numbers are the multiples of the factorials, 1.2.3, 220.127.116.11, 18.104.22.168.5, and so on. Thus we may regard
the multiples of six as highly divisible and prim, the multiples of 24 as even more highly divisible and more prim, and so on with
the multiples of 120, 720, 5040, etc. My observation is that the primes are deprived of factors by prims.
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 at the ‘prim’ numbers: those divisible by a high proportion of inferior integers, including 1, 2 and 3 (for small n, adding further factors according to the factorial series as n increases).
(see Figure 3).
AS/SA nº 14,
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 do indeed occur at every multiple of six (as well as every multiple of 24, 120 etc., obviously)
(represented here as yellow), and more interestingly, (pink) primes only occur at n ±1, or 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’
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 n that is evenly divisible by f must prim occur at nprim -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, nprim +1 and nprim -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.
AS/SA nº 14,
A Modulus-6 Clock Spiral
The prim number 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 4).