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:
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.
Write:
This notation may be confusing, so we provide an example to explain the first two rows of the table.
For the first row:
, so we are interested in the first prime.
is the first prime, namely 2.
is the second prime, namely 3.
is the third prime, namely 5.
is the fifth prime, namely 11.
is the eleventh prime, namely 31.
- etc.
We can rewrite this as:
The second row can be written as:
Things to note:
- Every prime appears in the table.
- The rows of the table constitute a partition of the natural numbers, with the first element being the only non-prime.
- 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:
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
- 179 is prime and is the 41st prime, i.e.
.
- 41 is the 13th prime, i.e.
, and also
.
- 13 is the 6th prime, i.e.
, and also
.
Thus, not only is 179 prime, but:
- The index of 179 amongst primes (41) is also prime.
- The index of 41 amongst primes (13) is also prime.
- 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:
- If
, the number is non-prime (aka composite) and
.
- If
, the number
is prime.
- 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:
- The first column consists of all non-prime numbers.
- For
, the
th column consists of all prime numbers of primality
.
- There are infinite numbers of primality
.
The rows partition the natural numbers into chains where, for a given chain :
we have the following properties:
- 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
.
- Every non-prime
generates a unique, infinitely long chain
.
- For any two non-primes
and
with
, we have that
.
- 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.