Note: Due to the inclusion of LaTeX in this file, pieces of text may appear to be missing in dark mode.

We begin by countably listing all primes and labelling them:

such that, for all :

1. is prime.
2. .
3. There is no prime such that .

Here is a list of the first 1000 primes in order for convenience:

https://en.wikipedia.org/wiki/List_of_prime_numbers#The_first_1000_prime_numbers

We create a partial segment of a table as follows. The ordering will be explained below.

1	2	3	5	11	31
4	7	17	59	277	1787
6	13	41	179	1063	...
8	19	67	331	2221	...
9	23	83	431	3001	...
10	29	109	461	3259	...
12	37	157	919	7193	...
14	43	191	1153	...	...
15	47	211	1297	...	...
16	53	241	1523	...	...
18	61	283	1847	...	...
20	71	353	2381	...	...
21	73	359	2417	...	...
22	79	401	2749	...	...
24	89	461	3259	...	...
25	97	509	3637	...	...
26	101	547	3943	...	...
27	103	563	4091	...	...
28	107	587	4273	...	...
30	127	709	5381	...	...
32	131	739	5623	...	...
33	137	773	5869	...	...
34	139	797	6113	...	...
35	149	859	6661	...	...
36	151	877	...	...	...
38	163	967	...	...	...
39	167	991	...	...	...
40	173	1031	...	...	...
42	181	1087	...	...	...
44	193	1171	...	...	...
45	197	1201	...	...	...
46	199	1217	...	...	...
48	223	1409	...	...	...
49	227	1433	...	...	...
50	229	1447	...	...	...

Write:

This notation may be confusing, so we provide an example to explain the first two rows of the table.

For the first row:

1. , so we are interested in the first prime.
2. is the first prime, namely 2.
3. is the second prime, namely 3.
4. is the third prime, namely 5.
5. is the fifth prime, namely 11.
6. is the eleventh prime, namely 31.
7. etc.

We can rewrite this as:

1. 
2. 
3. 
4. 
5. 
6. 
7. etc.

The second row can be written as:

1. 
2. 
3. 
4. 
5. 
6. 
7. etc.

Things to note:

1. Every prime appears in the table.
2. The rows of the table constitute a partition of the natural numbers, with the first element being the only non-prime.
3. The columns of the table also constitute a partition of the natural numbers, with the first column comprising the non-prime numbers.

Consider a prime number and let be a non-prime number where there exists a such that:

Then we say that has primality .

For a given prime in Definition 1.1, the numbers and are always unique.

Example: 17

Let's consider number 17. By our table, we have that:

Thus, 17 has primality 2 and falls in the row-partition of our table containing the non-prime number 4.

The significance of primality is as follows:

1. 17 is prime and is the 7th prime, i.e. .
2. 7 is prime and is the 4th prime, i.e. .

So. not only is 17 prime, but when we list all of the primes, the index of 17 - namely 7 - is also prime.

In this sense, 17 can be viewed as "more prime" than 7, which only has primality 1.

Another Example: 179

1. 179 is prime and is the 41st prime, i.e. .
2. 41 is the 13th prime, i.e. , and also .
3. 13 is the 6th prime, i.e. , and also .

Thus, not only is 179 prime, but:

1. The index of 179 amongst primes (41) is also prime.
2. The index of 41 amongst primes (13) is also prime.
3. The index of 13 amongst primes (6) is NOT prime.

In this case, we say that 179 has primality 3.

Let be a natural number such that, by Theorem 1.2, we have and such that:

Then we have that:

1. If , the number is non-prime (aka composite) and .
2. If , the number is prime.
3. If , the number is a higher-order prime (aka super-prime).

As mentioned, the table uniquely partitions the natural numbers in two different ways, namely via the rows and via the columns.

The columns partition the natural numbers according to their primality:

1. The first column consists of all non-prime numbers.
2. For , the th column consists of all prime numbers of primality .
3. There are infinite numbers of primality .

The rows partition the natural numbers into chains where, for a given chain :

we have the following properties:

1. The first cell in the chain is the unique non-prime in the chain, and thus we say that generates the chain, as you can see by the above expansion of . Thus, to uniquely identify , we refer to as .
2. Every non-prime generates a unique, infinitely long chain .
3. For any two non-primes and with , we have that .
4. For any chain , there is exactly one number of primality in , namely .

This repository is just some casual information about super-primality and some programs that assist in studying super-primality.