Factorizer 4. Palmen Colors
5. Plotting the Palmen Color Frequencies
6. Plotting the Prime Factor Frequencies
7. Plotting the Erdös-Kac histograms
8. Saving graph images
4. Palmen ColorsFactorizer also computes the Palmen color of a number. There are eight Palmen colors: black, red, green, yellow, blue, magenta, cyan and white. Each natural number has a unique color, and the color of a product of two numbers is determined by the colors of the two numbers. The eight colors in fact form a group. For further information about Palmen colors see Karl Palmen's Colours of Numbers.
The following are samples of the output obtained when the Show Palmen Color box is checked:
It seems that any pair of primes consists of a blue and a black number or a red and a green number.
5. Plotting the Palmen Color FrequenciesSome Palmen colors occur more frequently than others. Black is the most common and white is the least common. What of the other six Palmen colors? The Factorizer software allows us to explore their relative frequencies. For example, considering only the first 225 positive integers we obtain:
It is possible to use a filter when plotting Palmen color frequencies. This allows the experimental discovery of propositions about Palmen colors which are likely to be true although need to be proved mathematically. For example, with the filter 12*N - 1 and the range of integers from 1 through 1,000,000 we obtain:
which, of course, strongly suggests that all positive integers of the form 12*N - 1 are either yellow or blue.
The proportion of yellow numbers increases very slowly (41.63% by 20,000,000, 41.97% by 63,000,000). The proportion seems to be given by the expression 36.785*N0.0074, implying that it reaches 50% for N = 1018. Of course, this expression cannot be true for indefinitely large N, since in that case the proportion would exceed 100% for N > 1059. Clearly there is an upper bound (and so an exact limit) to the proportion of yellow numbers (of the form 12*N - 1) as N approaches infinity. Is it 50% exactly? Or does it perhaps exceed 50%?
6. Plotting the Prime Factor FrequenciesFactorizer illustrates this statistical property of the prime numbers in two ways. The first is simply by counting how many positive integers (over a certain range) have exactly 1, 2, 3 ... prime factors. For example, when we consider the 29 integers from 2 through 30 we find that there are 16 which have exactly one prime factor, namely, 2, 3, 4 = 22, 5, 7, 8 = 23, 9 = 32, 11, 13, 16 = 24, 17, 19, 23, 25 = 52, 27 = 33 and 29; 12 which have exactly two prime factors, namely, 6 = 2*3, 10 = 2*5, 12 = 22*3, 14 = 2*7, 15 = 3*5, 18 = 2*32, 20 =22*5, 21 = 3*7, 22 = 2*11, 24 = 23*3, 26 = 2*13 and 28 = 22*7; and 1 which has exactly three prime factors, namely 30 = 2*3*5.
Consider, for example, the range of integers from 2 through 32,000. We can use Factorizer to plot a histogram of the numbers of integers in this range, obtaining:
Clearly the counts tend to lie on the Gaussian curve defined by their mean and standard deviation.
If the software is run for many hours the mean increases gradually to beyond 3.147, suggesting that the quantity
where NPF(i) is the number of prime factors of i, has no limit.sum(i=2:n)NPF(i) ---------------- n - 1
7. Plotting the Erdös-Kac histogramsThe starting point here is the web page by Matthew Watkins entitled the Erdös-Kac theorem, where we read that the numbers of prime factors of large integers (suitably normalised) tend to follow the Gaussian distribution. This web page should be read for a deeper understanding of the significance of the Erdös-Kac histograms produced by the Factorizer software.
Matthew Watkins speaks of the numbers of prime factors of large integers "suitably normalised". By this he refers to the values, here called "EK values", defined as:
where NPF(n) is the number of prime factors of n, log() is the natural logarithm and sqrt() is the square root.NPF(n) - log(log(n)) -------------------- sqrt(log(log(n)))
Suppose we consider a range of integers and calculate the EK value for each integer. The range of EK values for numbers in the range 3 through 231 is -1.180 through 3.515 (the EK value for 2 is undefined), so we can divide this range of EK values into k "bins" and assign the calculated values to their appropriate bins, then count how many values are contained in each bin and plot a histogram. The Factorizer software will do this for us, and we obtain results such as:
The red Gaussian curve is defined by the mean and standard deviation of the calculated EK values, whereas the magenta Gaussian curve has a mean of 1/6 and a standard deviation of 2/3 (this seems to be, in some sense, the "ideal" for the distribution of EK values).
The Erdös-Kac histograms can also be plotted with a filter in effect. Curiously, the Gaussian property persists even with filters. For example, using the filter 12*N - 1 and plotting the integers with n prime factors over the range 10,000,000 to 10,100,000 we obtain:
How are these observations to be explained?
8. Saving graph imagesWhen plotting EK values or Palmen color frequencies the graphical image may be saved to a .bmp file by means of a right mouse-click (when the cursor is over the graph). This will bring up a dialog box asking if you wish to save the image. You can then select the name and location for the .bmp file in which the image is to be saved:
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