Prime FactorsPlotting the Erdös-Kac Histograms The 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 thePrime Factorssoftware.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) - log(log(n)) -------------------- sqrt(log(log(n)))NPF(n)is the number of prime factors ofn,log()is the natural logarithm andsqrt()is the square root.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 2

^{31}is -1.180 through 3.515 (the EK value for 2 is undefined), so we can divide this range of EK values intok"bins" and assign the calculated values to their appropriate bins, then count how many values are contained in each bin and plot a histogram. ThePrime Factorssoftware will do this for us, and we obtain results such as:

For more information see the page on the Erdös–Kac theorem at Wikipedia.

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