User:Robert A West/sandbox
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.
Sacks describes product curves as representing "products of factors with a fixed difference between them." Curves can be almost straight but, more typically, perform partial or complete revolutions near the origin. The "curviness" appears to be rooted in mathematical axioms - and is not an artificial construct. Expressed simply, the first complete rotation of the spiral necessarily comprises 1, 2, and 3 (since 1 and 4 are perfect squares) - from which multiple major curves originate. However, the spiral, taken as a whole, is aligned along four fundamental axes. This "dissonance" is formalized in the Offset Rule (see below).
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 an axis of pronic numbers, with the progression 2, 6, 12, 20, 30, 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), 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]."
An open question and a matter of particular interest and significance is the extent to which the number spiral's prime and factor curves are predictive of primes and composites far out in the number sphere.
Offset rule[edit]
Sacks' offset rule states: "A single rule determines the number of rotations for both even and odd curves: the number of revolutions is equal to the average of the offset's squarest integral factors. For example, the curve with offset 12 makes 3.5 rotations, and 3.5 is the average of 3 and 4, the squarest factors of 12."