This web page is an expanded version of a presentation given in Boston at the 7th International Conference on Complex Systems

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Prime Numbers are the Holes Behind Complex Composite Patterns
Visualizing Number Theory with the Divisor Plot

JJ Ventrella
Jeffrey@Ventrella.com
www.Ventrella.com

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3 The Square Root Boundary

One way to determine whether an integer x is composite is to test whether it is evenly divisible by any integer i (1 < i <= sqrt(x)). It is not necessary to test for larger i since all divisors come in conjugate pairs (i1 < sqrt(x), and i2 > sqrt(x)) except for the roots of perfect squares, in which case i = sqrt(x). This dual property of divisors is expressed in the region of the curve y = sqrt(x). The image below shows the beginning of this curve.

The square roots of the first six perfect squares, (1, 4, 9, 16, 25, 36) are highlighted with circles. This boundary becomes increasingly horizontal at higher numbers. The image below at right shows x = 100 and its square root 10, lying on the boundary. The green curved sweep of ordinal numbers that includes 10 is one of the square root parabolas, explained below.

Wolfram describes a quadradic sieve that can generate the prime numbers, using the integer points lying on the parabola x = y2. Connect each pair of points occurring above and below the x-axis. The points where these lines intersect the x-axis correspond to composite numbers. Points where no lines cross are prime numbers. The bottom arm of the parabola lies on the square root boundary. As yet I do not know the significance of this, if there is any.

Symmetry The image below shows D36. Its pairs are (1, 36) (2, 18) (3, 12) (4, 9). Its square root is 6. Notice that the two numbers in each pair in this series converge towards the square root. This is a hint of the kind of symmetry that becomes more complex at larger numbers.

For instance, the image below shows the square root boundary at one million, with its square root 1000 indicated by the arrow.