Euler's Product formula upside down

This model shows how if you put $1, 2, 2^2, 2^3, ...$, along one axis, $1,3, 3^2, 3^3, ...$ along another and $1, 5, 5^2, 5^3, ...$ along a third and then fill out the 3D table by multiplying everything, you end up getting all possible numbers made from 2s, 3s and 5s.

If you imagine this with a dimension for every prime number you get the formula $\prod{\sum{p^k}} = \sum{n}$

Pascal's triangle

Here is a Lego model of Pascal's Triangle.

Each number in the triangle is represented by a tower of its prime factors. By changing the order of the blocks you can make the divisibility properties easy to see. Here is the model with all the orange 7 blocks moved to the top.

The upside-down orange triangle shows the numbers that are divisible by 7.

Factorials

This is the sequence of factorials. Each factorial is made by putting it and all the numbers before it into one big tower. Here is the number line from 1 to 9.

By looking at the colours one at a time, you can work out how many blocks of each colour there is in a factorial. Here are only the blue blocks in the number line from 1 to 20

The number of blue blocks in n factorial is

$\lfloor \frac{n!}{2} \rfloor + \lfloor \frac{n!}{4} \rfloor + \lfloor \frac{n!}{8} \rfloor + \ldots$

Fiddling with Euclid's Proof

When you add relatively prime numbers you get an answer which is relatively prime to each of the starting numbers.  This means in Lego you get all new colours. We can use this to prove that there are infinitely many primes.  It is very similar to Euclid's proof, but it 'generates' more primes.

You start by assuming there are only finitely many primes - in Lego take one block of each colour. Then make two towers and add them together. The answer must always be made out of new colours.

Here we start with the blocks for 2, 3, & 5, and end up with new blocks for 17, 13, & 11.

Addition, Multiples, & Relatively Prime Numbers

When you add two numbers in Lego you keep the colours that are shared and drop the colours that are not shared. This works because when you add two multiples of any number, you get another multiple of that number.

$an+bn=(a+b)n$

Adding two even numbers gives another even number.  These are the multiples of 2: all the numbers which contain a blue block.

Numbers are relatively prime if they don't share any prime factors.  In Lego this means that numbers are  relatively prime if they don't share any colours.

Pell numbers

This is the sequence of Pell numbers made from Lego.

Again the colours repeat in regular patterns.

$P_2 | P_2, P_4, P_8, \ldots$

$P_3 | P_3, P_6, P_9, \ldots$

$P_4 | P_4, P_8, P_{12}, \ldots$

$P_5 | P_5, P_{10}, P_{15}, \ldots$

So this is almost exactly like the counting numbers. The 'P-primes' are P₂,P₃, P₅, P₇, ...

Fibonacci Sequence

Here is the Fibonacci sequence made in Lego in the same way.

Just as in the number line you can see how the colours repeat in a regular pattern. In the Fibonacci sequence the blue block repeats every third time, and the red block every fourth time, and so on.

So we can see that
$2 | F_3, F_6, F_9, \ldots$
$3 | F_4, F_8, F_{12}, \ldots$
$5 | F_5, F_{10}, F_{15}, \ldots$

Or another way to write it is

$F_3 | F_3, F_6, F_9, \ldots$

$F_4 | F_4, F_8, F_{12}, \ldots$

$F_5 | F_5, F_{10}, F_{15}, \ldots$

So the 'F' numbers make a number line just like the counting numbers, except that F₂ is not 'prime' ... this means that F₄ has to be an 'F-prime'.