EDIT: Because of what happened in the first page, this is slightly different.

As you may know, every number can be prime-factorized. For instance,

21 = 3*7,

24 = 2

^{3}*3 = 2*2*2*3.

Now, 3 is the second prime -- we can represent 3 as p<2>, or p2. Similarly, 7 is the fourth, so we note it p4. Thus, 21 is p2p4; we use concatenation to multiply.

I called this recursive prime factorization: 2 is the first prime, so we can represent 3 as p<p<1>>. By doing it this way, we reduce all numbers into a series of brackets, down to the "root" 1. We don't actually need that; we could just as well represent that as nothing.

This is the basic idea; the goal in this thread is to post the next number using this format in some way. Some common formats are:

Omit the "p": 3 becomes <<>>. This is bracket notation/b-notation

Leave the "p", but omit unnecessary brackets (say, for composite-numbered primes.) 6 is p1pp1, 7 is p<p1p1>. We call this p-notation. This was the original one used for this thread.

Represent it as a tree: There are many ways to do this. Examples are in the spoilers below.

Lineal-Spatial (L-S) Notation

**Spoiler:**

Depth Notation

**Spoiler:**

There are, of course, tons of possible notations; people keep making more. Again, as long as you show this recursive structure somehow, it is fine.

Alternatively, you could use one of the many common shorthands for some of the common numbers, as seen below.

As examples, I will give the numbers from 1 to 10 using bracket notation, p-notation, and decimal. Consensus has built to use bracket notation, with some random other notations, although I strongly suggest including decimal as well. I also include some common shorthand (namely: circle-notation, abc-notation, and dot-notation. These are often mixed.).

, 1, (1)

<>, p1 (2) (Shorthand: . (dot), o (circle), a (abc/alphabetical), (fancy circle))

(<>), pp1 (3) (Shorthand: ⊙ (circle), b (abc/alphabetical), ۞ (fancy circle))

<><>, p1p1 (4) (Shorthand: : (dot))

<[<>]>, ppp1 (5) (Shorthand: c (abc/alphabetical)

<><[]>, p1pp1 (6)

<[][]>, p<p1p1> (7) (Shorthand: (:) (dot))

<>[]<>, p1p1p1 (8)

<[]><[]>, pp1pp1 (9)

<><[<>]>, p1ppp1 (10).

Let us continue.

<<<<>>>>, pppp1 (11). (Shorthand: d (abc/alphabetical))