Introduction
The Spiral Plane
Product Curves and Polynomials

Introduction

In 1994, Robert Sacks, a software engineer, devised an original method for representing the classical number line of positive integers. He published his findings on the web in 2003. In this method, an Archimedean spiral centered on zero and making one counterclockwise rotation for each perfect square produces a remarkably organized distribution of prime and composite numbers. In some respects, this 2-dimensional "number sphere" can be thought of as a periodic table of numbers - because of the orderly patterns and progressions it reveals.

The Sacks number spiral is both visually arresting and intellectually compelling. It seems likely that it can provide deeper insights into prime number patterns than the well-known Ulam spiral because, in effect, it joins together the broken lines of Stanislaw Ulam's pseudo-spiral.

Sacks' work has focused on product curves, lines that originate from the spiral's center, or near to it - and traverse the spiral's arms at varying angles. His graphing demonstrates that there are multiple orderly factor and prime number progressions. Since the spiral can be extended outward infinitely, the product curves themselves may be considered infinite also.

It's an open question whether or not the spiral has any predictive value. For example, can the number spiral's product curves be used to predict the decomposition of very large numbers into prime factors?

The Spiral Plane

Like the Cartesian plane and the Complex plane, the number spiral represents a plane in which points can occupy one location only. Unlike the traditional planes, all numbers are natural numbers: the positive integers. (However, a spiral of negative integers with negative perfect squares can be visualized as sharing the zero-point center and rotating clockwise in a parallel plane. This is conceptually useful because many polynomial sequences begin with negative integers, then enter the positive spiral, before returning to the negative region.)

Quite unlike the Cartesian and Complex planes, the Spiral plane does not require a coordinate system to identify a point. It is a unipolar positioning system. Just knowing the number reveals its location in the spiral - its position relative to every other number in the spiral, its distance from the previous and the next perfect squares, and, potentially, all the curves it can occupy relative to other numbers in the spiral: its polynomial classification.

The parabolic curve is a well-known attribute of polynomial equations in classical algebra. In fact, solving the roots of a polynomial means finding the points along the X axis of the Cartesian plane that the parabola intersects. So curviness appears to be rooted in mathematical axioms, but is shown in a new light by the Spiral plane. The fact that the roots of many polynomials, even simple ones, cannot be solved in the Cartesian plane – that is, they do not intersect with the X axis – is one of the intellectual justifications for the Complex plane and Complex numbers.

Underlying the distribution of numbers in the spiral plane is the alignment of perfect squares. The progression of perfect squares determines the expansion of the spiral. The first complete rotation of the spiral necessarily comprises three poles - 1, 2, and 3 (a perfect square and two primes). However, the spiral, taken as a whole, aligns along four axes, which can be designated, for convenience, as north, south, west, and east.

Zero degrees (aligned due east in the Sacks spiral) is the primary axis and corresponds with the perfect squares: 1 (1*1), 4 (2*2), 9 (3*3), 16 (4*4), 25 (5*5), and so on. Due west is the axis of pronic numbers, with the progression 2 (1*1+1), 6 (2*2+2), 12 (3*3+3), 20 (4*4+4), 30 (5*5+5), and so on. Due north and south are predominantly composite numbers (beyond the second rotation) and follow relatively more intricate progressions, spawning multiple east-west product curves.

One of the most striking aspects of the Sacks number spiral is the predominance of major prime curves in the western hemisphere (opposing side from the perfect squares). For example, one of those curves, heading south-west, contains the numbers of the form n(n + 1) + 41, which is a famous prime-generating formula discovered by Leonhard Euler in 1774. In the number spiral, Sacks is able to make the striking assertion that a prime number is "a positive integer that lies on only one product curve."

Focusing on the pronic curve (which is almost a straight line), forming the western axis, Sacks states that "we can continue adding product curves indefinitely by increasing the difference between factors.... Odd curves (those with factors that differ by odd numbers) are offset in the clockwise (minus) direction [from the pronic curve], and even curves are offset in the same [clockwise direction but from the perfect squares]."

Product Curves and Polynomials

Sacks describes product curves as representing "products of factors with a fixed difference between them". In other words, every curve can be represented by a quadratic equation - a second-degree polynomial - no coincidence given the primacy of the perfect square in the structure of the spiral.

Curves can be almost straight but, more typically, perform partial, complete, or multiple clockwise revolutions - counter to the spiral itself - around the origin before straightening out at a particular offset from the east-west axis. The number of revolutions is the direct product of the offset. The greater the offset, the greater the number of revolutions a curve must make before pulling free from its initial "orbit". This relationship is formalized in the Sacks Offset Rule.

Product curves can contain composites or primes, the relative proportion of each being predicated on the all-important offset. Simple observation shows that curves with prime number offsets are reliably richer in primes than other odd-number composite offsets. This observation is the basis for the work on this site related to proximate-prime polynomials.

Why do the curves straighten out anyway? It turns out the answer is quite simple and has to do with how polynomials are related to perfect squares. A polynomial curve straightens out when its values align with the median values of successive perfect squares. This axis comprises the pronic numbers, which, as previously stated, can be expressed as n² + n. For more about this, see Polynomial Pronic Alignment. The pronic curve is, in fact, the simplest of all 2nd-degree polynomials - one without a third coefficient, or offset.

The point at which a polynomial curve aligns with the pronic axis is entirely predicated on the third coefficient of a 2nd-degree polynomial equation. A product curve for a given polynomial (of type n² +/- n + x, where x is the offset) travels clockwise, counter to the spiral itself, before intersecting with the squared value of the offset number on the spiral. This is the point of pronic alignment. An offset of 41 literally means the curve will align with the pronic axis at 41*41 - the composite number 1681. From that point forward, the curve is aligned with the pronic axis.

Is the curvy portion of a curve different somehow than the (infinite) straight portion? The answer to this question is not clear. It appears plausible that the greatest concentration of primes for many polynomials occurs within the pre-pronic-alignment portion of their paths across the number spiral. This is dramatically illustrated by the uninterrupted prime sequence of a perfect prime polynomial, such as n²+n+41. But it's not proven as a general rule.

Michael M. Ross