Tuesday, 10 January 2012

Fractal patterns in composite n-factor numbers

Here's something I find weirdly fascinating, quite likely better understood by proper mathematicians... In a previous post I observed that

"The pattern of 6N+/-1 primes eg 1 x 5, 1 x 7, 1 x 11 etc is also fractally replicated in n-factor numbers. For 2-factor numbers a new pattern (congruent to the overall pattern of primes) starts at each prime  eg 5 x 5, 5 x 7, 5 x 11…  7 x 5, 7 x 7, 7 x 11…. and so on. For 3-factor numbers we have a new repeat of the prime pattern starting from each 2-factor number and so on."

Now, we can tell a few things about these overlaying patterns. Firstly, we know that each iteration of the pattern is congruent (or is "isometric" a better word??) to the original prime pattern albeit proportionately bigger - so, 5, 7, 11, 13, 17 is congruent to 25, 35, 55, 65, 85, which is congruent to 35, 49, 77, 91, 119 with each entry in the latter being 7/5 bigger than the second, and 7 times bigger than the first. the first 3-factor iterations of the pattern are 125, 175, 275, 325, 425... then you get 175, 245, 385, 455, 595 etc.

Secondly, we know that as we overlay the n-factor patterns, each number can only appear n times - in other words, 35 can appear twice in the lists above, but only twice as it only has two factors. (25 can only appear once as it is a square, so the key thing is how many distinct factors a number has, not how many factors).

Next, and this is something I find intriguing, we know that each iteration of these congruent patterns picks out a unique set of numbers - this means that there is no number that is in both the 2-factor pattern and the 3-factor one. So, to take a visual approach, the set of composite numbers of the form 6N+/-1 is made up of infinite iterations of the same original infinite pattern of primes, all delicately avoiding landing on the same number off to infinity.

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