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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4 Parabolas

If you look closely at images of the divisor plot, you might be able to make out some parabolas, by connecting the dots. These are like the cross-sections of compressed, folded pieces of lettuce in a Dagwood sandwich. The most distinct parabolas are from a series that I call "square root parabolas" - an infinite set of nested parabolas P whose vertices run along the square root boundary. They are numbered n from 1 to infinity. The equation for a square root parabola Pn is: x = (n/2)2 - (y - n/2 )2 ...where y ranges from 0 to n. Consider this as an exponential diophantine equation, which defines a set of lattice points that correspond to sets of divisors, numbered 1 to n, as shown at right.

The entire set P fully tessellates the divisor plot - it intersects every divisor. The image below shows P1 through P8, rendered with straight lines connecting the ordinal series of divisors that lie on them. The arms of parabola Pn intersect (0,0) and (0, n). Even-numbered Pn have perfect square root divisors located at their vertices (highlighted in red in the image). Odd-numbered Pn have pairs of divisors with a difference of 1 (making them divisors of pronic numbers) near their vertices. These are highlighted in green.

All Pn have the same overall shape. But due to their x locations on the lattice, even and odd-numbered Pn exhibit different spacings in x between their divisors.

A Conjecture about Squares Fibonacci (and I'm sure many others) have observed that square numbers can be constructed as sums of odd numbers. Notice in the even-numbered parabola above that the x-spacing between divisor pairs corresponds to the series of odd numbers, counting from right to left. Suppose that the vertex of this parabola is located at (100, 10). If you work backwards from the vertex, and add the odd numbers shown, you get a series of squares: 1, 4, 9, 16, 25, 36, etc.

Now, subtract each of these square numbers from 100. Subtract 1 from 100, and you get 99, which is the x value for the pair immediately to the left of the vertex. Subtract 4, and you get 96, which is the x value for the pair immediately to the left of that. Subtract 9 and you get 91... And so-on. Subtracting each of these square numbers from the x value of the vertex results in the x values for each pair of divisors on the parabola. Since we know that P represents a set of composite numbers, we arrive at the following conjecture:

Take any square number x, and then take the set of all square numbers less than x and subtract them from x. All the numbers in this new set (except the lowest) are composite.

Constant Sums Rob Sacks observed that all divisor pairs in a square root parabola Pn have the same sum. In fact they are equal to n. This leads to another definition of a square root parabola: "a square root parabola n is the curve which intersects all divisor pairs whose sums equal n."

Divisor Pairs on Sqrt Parabolas The image at right shows divisors near the vertex of P128, whose vertex is at (4096, 64). Notice how divisor pairs to the left of the vertex converge towards the square root boundary as they get closer, and that there are no divisor pairs immediately to the right. Why is this?

The image at left shows P2 through P8. Below them are symbols representing the products of the divisor pairs on these parabolas. Each symbol is a rectangular array of blocks, where the number of blocks is equal to the product. The number of columns in the array equals the divisor on the top arm of the parabola, and the number of rows equals the divisor on the bottom arm of the parabola (with the exception of divisors at the vertex - the square roots). For example, the number 12 is shown as occurring on P7 and P8. The associated symbols show two of the three ways to express 12 (2x6 and 3x4). (The third way, 1x12, occurs on P13, which is not included in this illustration). This illustration may help give insight as to why these parabolas exist. The answer may have to do with the possible ways to arrange these blocks into rectangular arrays. Note that primes can only be represented in single-column, and that their divisors are always the second pair occurring at the left edge of a parabola (after the pair occurring at x=0).

Approaching "Squareness" at Infinity

In Pn at increasingly higher x, the divisor pairs that are near the vertices converge towards a ratio of 1. And the divisor pairs that are farther away from P vertices converge more slowly towards a ratio of 1, as x increases. Since all Pn have the same shape, numbers which are one-less than perfect squares converge towards "squareness", as x increases. For instance, compare the numbers 35 and 18,768, both of which are followed by a square number. Their closest divisors pairs are (5,7), and (136,138), respectively.

Even though each pair has a difference of 2, the ratios of those pairs are 0.714 and 0.985. The ratios of pairs of divisors preceding P vertices converge to 1 as x approaches infinity. All divisor pairs are members of a square root parabola, and all square root parabolas are continually becoming more "compressed" as x increases. In other words, the average distance between divisor pairs becomes proportionally smaller at higher x, relative to the range of divisors as a whole. Is there something "attractive" about the square root boundary?

Other Parabola Series Other parabola series exist, in addition to those at the square root boundary – they become more apparent as we travel to larger numbers. The images below show parabolas above and below the square root boundary in a window with the y range scaled 3 times the x range to make them easier to see. The square root boundary is indicated with a line. The left image shows a pair of divisors located at the vertices of parabolas at x = 8192 = 213. The middle image shows a pair of divisors located at the vertices of parabolas at x = 16384 = 214, plus its square root.

Power of two x = 2n appear to be associated with parabola vertices. Mersenne primes (primes of the form 2n-1) have been studied throughout history. Patterns in the gaps in x immediately to the left of these vertices (such as the Mersenne prime 213-1) may provide insight. The right image shows divisors of 100,000 close to the square root boundary. In this case the parabolas are emphasized with overlaid lines. Note that each divisor is located on the opposite side of the parabola in relation to the other divisor in the conjugate pair.

Complimentary Parabolas We have just seen several examples of parabolas which come in pairs - analogous to pairs of divisors on opposite arms of a square root parabola. What can we learn about composite numbers by studying these parabolas? In the image at right, we see a pair of parabolas lying above and below the square root boundary. The number 10366 is equal to 71 times 146, and also 73 times 142. Is there any significance to the fact that these pairs are so close? And what is the significance of 10368 having a pair of divisors each lying at parabola vertices? Might 10368 have something in common with the perfect squares or the pronics?

2n Parabolas We have looked at a few examples of parabolas whose vertices lie at power of two (2n) x values. Notice that their y values are also 2n numbers. It turns out that there are many curves in the divisor plot which contain 2n parabolas, including the square root boundary itself (they are located at x = 2n, where n is even, as we saw above in the example of 214). The image below at left shows a schematic of the divisor plot. Both the x and y dimensions are scaled exponentially, such that x = 2n, and y = 2k. The values of n and k are integers, represented by the lines in the graph.

Notice that the square root boundary (red line) appears straight as a result of this scaling. On either side of the square root boundary are white lines where 2n parabola vertices lie (specifically, at the locations where the graph lines cross). Two examples are shown at right, indicated by light-blue dots. Notice that the white lines start off transparent and get brighter towards the lower-right. They were rendered this way because it is difficult to detect 2n parabolas at smaller x, or far from the square root boundary. This is not to say that they don't exist - just that I haven't identified any.

The diagonal lines in this graph follow curves of the equation....

y = 2log2(x)/2 + a

... where a ranges from -2 to 2 in increments of 0.5. If you look closely in the image above at right you can make out a few parabola series that lie at y values in-between the parabola series represented in the graph. It appears that all parabola series follow curves defined by this equation, where a can be many different values. When a is zero, we get the square root boundary. Perhaps the values of a might give us some insight into why these parabola series exist.

Parabolas Galore There are many more parabola series to explore besides those described here. And they exhibit different properties. For instance, parabolas appear to have variations in scale along y - they are "fatter" towards the bottom, at larger y. Also, their divisors are ordered differently along the curve. They may not always line up vertically, as in the case of the square root series, and the values of divisors, running clockwise along the parabola sometimes skip by two's, three's, and so-on.

Is there significance to the fact that x at 2n, perfect squares, and at D100000 have associated parabola vertices? What other kinds of x are associated with vertices? Another question: could it be that ALL parabola series tessellate the entire domain, as we have noted with square root parabolas? If so, perhaps composite numbers might be described in terms of all the parabola series that contain their divisors. It seems that every question I come up with creates 10 more questions!