![]() ![]() The closest together that two larger primes can get still keeps them separated by at least a single even number. For this reason there are no consecutive primes other than two and three. Other than two, all prime numbers are odd. Prime powers produce many familiar fractal constructions in their factorization diagrams, including Cantor dust and the Sierpinski gasket. ![]() This is a well known-way of creating fractals. The process of constructing the factorization diagram will mean repeated iteration of the same procedure, over and over, at many different scales. Below I have listed some interesting configurations to look out for.Ĭonsider a number that is a prime raised to a power (for example 256, which is 2 8). It's quite mesmerizing to watch the evolving factorization diagrams. Zooming back out we have created an arrangement of thirty dots: pairs, clustered into triples, which are then clustered into a ring of five. Finally, we subdivide each of those into two more circles. We then subdivide each circle into three smaller circles. Thirty dots, for instance, factors into 5 x 3 x 2. You can think of the prime factors of a number as being like that number's fingerprint, unique and indelible.įor a given number of dots, the animation finds an arrangement that uses each of that number's prime factors as a visible symmetry. Every number has only one factorization (5 x 3 x 2 is the only set of primes whose product is thirty) and each number's is different. This process of unpacking a number into a set of primes is called prime factorization. Second, although we have arrived by different routes, we've ended up with the same collection of primes in all three of the above cases: 2 x 3 x 5 (in some order). Every number being multiplied is prime and since primes are indivisible we cannot break these products down any further. First, the process of factoring is now over. We can continue this game, breaking the numbers in the equations into still smaller factors.īut at this point a couple of interesting things have happened. If we wanted to "build" this number from the multiplication of smaller numbers, we would have several ways to do it: In this post, I would like to explain the process by which the dots get arranged and point out some things it reveals about the structure of the natural numbers. ![]() But the really interesting thing is the appearance of the non-prime numbers, the composites, which display a wild variety of different patterns. The prime quantities can be easily identified because they form simple rings. Stephen Von Worley, extending an idea of Brent Yorgey's called Factorization Diagrams, created an animation which adds one dot after another to the screen while systematically arranging them into different patterns. In a previous post I described a method of arranging a collection of dots in order to determine if their quanitity was prime. ![]()
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