Question about the gaps between prime numbers

[DuckAmuck]

Is there any prime number p_n, such that it has a relationship with the next prime number p_n+1
$$p_{n+1} > p_{n}^2$$
If not, is there any proof saying a prime like this does not exist?

I have the exact same question about this relation:
$$p_{n+1} > 2p_{n}$$

 

[fresh_42]

Is there any prime number p_n, such that it has a relationship with the next prime number p_n+1
$$p_{n+1} > p_{n}^2$$
If not, is there any proof saying a prime like this does not exist?

I have the exact same question about this relation:
$$p_{n+1} > 2p_{n}$$

https://en.wikipedia.org/wiki/Prime_gap

There is also a proof for arbitrary gaps, but see the section "upper bounds".

 

[Stephen Tashi]

I have the exact same question about this relation:
$$p_{n+1} > 2p_{n}$$

That would contradict Bertands Postulate https://en.wikipedia.org/wiki/Bertrand's_postulate

 

[DuckAmuck]

Interesting.Bertrand's Postulate answers the second part of my question. :)

I see Firoozbakht's conjecture, which is similar to my first part, but it's not quite the same thing as
$$p_{n+1} > p_{n}^2$$

I wonder if this can be proved or disproved from other postulates...

 

[micromass]

Interesting.Bertrand's Postulate answers the second part of my question. :)

I see Firoozbakht's conjecture, which is similar to my first part, but it's not quite the same thing as
$$p_{n+1} > p_{n}^2$$

I wonder if this can be proved or disproved from other postulates...

This also follows very easily from Bertrand's postulate.

 

[Stephen Tashi]

but it's not quite the same thing as
$$p_{n+1} > p_{n}^2$$

I wonder if this can be proved or disproved from other postulates...

Compare $2p_n$ to $p^2_n$ .

 

[DuckAmuck]

This also follows very easily from Bertrand's postulate.

Yeah it does. Wow. I'm dumb. :p