Who Says that Primes Have No Formula

Who says that primes

Have no formula

And that they are random

We all know that

Primes are in a set

And that set is P

Let

1^st prime be p₁

2^nd be p₂

3^rd be p₃

And then so on

We don’t care

What their values are

What we care

Is that they come in order

And that order is

p₁ is less than p₂

p₂ is less than p₃

And then so on

Dropping the symbol p

And mapping primes

Into the set of subscripts

Then

1^st prime is 1

2^nd is 2

3^rd is 3

And then so on

Did you notice

The formula is so simple

n^th prime would, therefore, be n

What!

If primes take the place of integers

Then where did the composites go

Very good friend!

If primes take the place of integers

Then there’s no place for composites in the set of integers

Just like there is no place for rationals in the set of integers

Here is how they go

If primes take the place of integers

Then the product of integers

Is a composite

Just like the ratio of integers

Is a rational

As primes take the place of integers

Let one integer be a

And the other be b

Then

a times b

Is a composite

Just like

a over b

Is a rational

What!

If we define

A larger set S

That contains

Primes

Composites

And rationals

Then on the number line

Where exactly do they lie?

Good friend!

Since primes take the place of integers

On the number line, therefore

composites lie between primes

Just like rational do

To conclude the discourse

There is no difference between the set of integers

And a set isomorphic to integers

This is how it happens

Let

a_i be an element of the isomorphic set

Dropping the symbol a

And leaving out the subscript i

Thus every isomorphic set

Reduces to the ground set