Most Abundant form of Prime Numbers

[CRGreathouse]

Originally Posted by Dr Sardonicus

AFAIK, the question of whether a given (monic, irreducible) polynomial in Z[x] without any intrinsic prime factors even assumes infinitely many prime values is unanswered for any polynomial of degree greater than 1.

I also think that's unanswered. This Green-Tao paper
https://arxiv.org/abs/math/0606088
I think is more-or-less state of the art in that area, even though it predates Zhang's theorem.

[a1call]

Unless I am missing something, any factorial can be written as high degree polynomial, n(n+1)(n+2),...
So unless there is a finite number of primes of the form n!(+-)p for p>n,
then there has to be unlimited primes of polynomial form of degrees higher than 1.

ETA I know I have to be misunderstanding what is being stated.

[VBCurtis]

Originally Posted by a1call

Unless I am missing something, any factorial can be written as high degree polynomial, n(n+1)(n+2),...
So unless there is a finite number of primes of the form n!(+-)p for p>n,
then there has to be unlimited primes of polynomial form of degrees higher than 1.

ETA I know I have to be misunderstanding what is being stated.

What does ETA stand for when you use it like this?

The statement is that ONE polynomial produces an infinite number of primes (over the domain of input numbers n). You're finding a different poly for each prime of some form, without considering what other primes that poly would produce.

[axn]

Originally Posted by VBCurtis

What does ETA stand for when you use it like this?

Edited to add

[Batalov]

"P.S." is a more common way to say the same ("an afterthought").

[MattcAnderson]

Don't forget that, except for 2 ,, and 3, all prime numbers have the form 6n plus or
Onus one

Regards
Matt

[science_man_88]

all primes p are p-1 rough numbers.