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Natural numbers greater than 1 are either **composite** numbers or they are **prime** numbers.
We can **decompose** a composite number into prime factors.
For example, the number 75 can be decomposed into 3 * 5 * 5,
where 3 and 5 are prime numbers.

Here is a function that decomposes a number into a list of its prime factors.

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Some numbers, like 2, or 97, are not composite numbers,
they cannot be further decomposed.
They are **prime numbers**.

For prime numbers, the "decomposition" is a list with just one element, the number itself.

In 2012 computer scientist Brent Yorgey published a
blog post
describing a nice way to visualize the prime decomposition of a number.
He called his visualization a **factorization diagram**.

For example, the number `75 = 3 * 5 * 5`

is represented as follows:

The first prime factor, 3, is represented by the green triangle. At each of the three corners we find a cyan pentagon (the second prime factor, 5), and at each of the corners of those pentagons, we again find a pentagon (the third prime factor, also a 5). At each corner of those pentagons we then find a black dot.

The total number of black dots is 3 times 5 times 5 = 75. The diagram represents the number 75 by recursively decomposing it into substructures.

Here is another diagram, for the number `1155 = 3 * 5 * 7 * 11`

:

At the top level we have a triangle, breaking the number into three equal components (each with a total of 1155 / 3 = 385 dots). At the core of each component is a pentagon, breaking that component down into 5 equal subcomponents (each with a total of 385 / 5 = 77 dots). Each of those subcomponents is a heptagon (a seven-sided polygon), further breaking it down into 7 equal subcomponents (each with a total of 77 / 7 = 11 dots). At the bottom of the decomposition we have hendecagons (eleven-sided polygons), breaking it down into 11 dots.

Prime numbers cannot be further decomposed (that's the whole point of prime numbers). Thus, their factorization diagrams consist of a single polygon.

Here are the first few prime numbers:

This activity has been created by LuCE Research Lab and is licensed under CC BY-SA 4.0.

Factorization Diagram

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